LIBRARY 
 
 University of California. 
 
 RECEIVED BY EXCHANGE 
 
 Class 
 
ON THE PENTACARDIOID 
 
 ABSTRACT OF A 
 
 DISSERTATION 
 
 Submitted to the Board of University Studies of the Johns Hopkins 
 
 University in conformity with the requirements for the 
 
 degree of Doctor of Philosophy 
 
 BY 
 
 EDWARD C. PHILLIPS, S. J. 
 
 March, 1908 
 
 
 BALTIMORE, MD., U. S. A. 
 1909 
 
ON THE PENTACARDIOID 
 
 ABSTRACT OF A 
 
 DISSERTATION 
 
 Submitted to the Board of University Studies of the Johns Hopkins 
 
 University in conformity with the requirements for the 
 
 degree of Doctor of Philosophy 
 
 BY 
 
 EDWARD C. PHILLIPS, S. J. 
 March, 1908 
 
 -■ OF 1 HE 
 
 UNIVERSITI. 
 
 OF 
 ^ALIFOf 
 
 BALTIMORE, MD., U. S. A. 
 1909 
 
/ 
 
 The writer desires to express his gratitude for the 
 constant interest and kindly encouragement and -assist- 
 ance given to him by Professor Morley, not only "during 
 the preparation of this paper but during his entire 
 course at the University. 
 
 
ON THE PENTACAliDIOID. 
 
 Bt Edward C. Phillips. S. J. 
 
 In the discussion of the metrical properties of finite systems of lines in a 
 plane there arises a series of curves called by Professor Morley Ennacardioids;* 
 these curves, over and above their usefulness in connection with the system of 
 lines from which they arise, have many interesting properties which seem 
 worthy of some special investigation ; and it is proposed in this article to make a 
 study of the Pentacardioid, the first in the series of Ennacardioids which has 
 not as yet received any detailed treatment. The general symbol for an Bnna- 
 cardioid is C", and we shall use throughout this paper the corresponding symbol 
 C^ for the Pehtacardioid. 
 
 The system of coordinates and the general method of analysis which will be 
 employed are those described in § 1 of Professor Morley's Memoir on Reflexive 
 Geometry. However, as the notation in the various articles on this and allied 
 subjects varies considerably, I shall, in order to avoid confusion, here state 
 briefly the symbolism I intend to follow. The conjugate of a complex number x 
 will be represented by the letter y, and the conjugate of a, by h. A complex 
 number of absolute value equal to unity is called a turn and will in general be 
 denoted by the letter < or by a Greek letter; the conjugate of a turn being its 
 reciprocal needs no special notation. There are certain special turns to which 
 definite symbols have already been assigned, and these I shall retain ; they are 
 the following : The square root of negative unity, designated by i ; the cube roots 
 of unity, designated by w and its powers ; and the fifth roots of unity, designated 
 by £ and its powers Thus these three symbols are defined by the equations 
 i^ + 1 = 0, o' — 1=0, and e^ — 1 = 0. The modulus, or absolute value, of 
 a complex number will be denoted by the letter p, or by placing the number 
 between parallel strokes, thus : | a; | . A turn may also be expressed exponentially, 
 
 *F. Morley: On Reflexive Geometry, Transactiont of the American Mathematical Society, Vol. VIII (1907), 
 pp. 15 ff. 
 
 186967 
 
2 ON THE PENTACARDIOID. 
 
 and thus any complex number is of the form ft, or pe**, where 6 is the amplitude 
 of the complex number. 
 
 An equation in complex variables is said to be self-conjugate when it is 
 identical with the equation obtained by replacing each quantity by its conjugate, 
 or differs from this equation merely by some factor; such an equation corre- 
 sponds to an equation in real variables with real coefficients. The roots of a 
 self-conjugate equation in t are either turns or pairs of inverse points as to the 
 base circle, and these latter bear to the self-conjugate equation in i the same 
 relation that pairs of conjugate imaginary roots bear to the equation with real 
 coefficients in one real variable. 
 
 § 1. The Equation of the Curve. 
 
 The Pentacardioid, or C"*, as defined by its line equation in its most general 
 form, is 
 
 X— ao -f ba^t -f IQa^^ + \Qh.f + bh^t^ + {V — hY = 0. (1) 
 
 '» 
 
 The turn t is here used as a parameter ; each value of t gives, for a varying x, a 
 
 line or tangent of the curve, and the envelope of this system of lines is the point 
 
 curve. The equation of the point curve can be obtained in map form by 
 
 eliminating «/ between equation (1) and its derivative as to t\ the elimination 
 
 gives 
 
 x = ao— 4ai<— 6a2<2— 462<» — 6i<*. (2) 
 
 The point Oq is the singular focus of the curve and will be called its center. 
 
 There are in the above equation three independent complex arbitFary con- 
 stants, and hence a C'° depends on six conditions. It takes two complex constants, 
 equivalent to four conditions, to fix the reference system so that there are only 
 two absolute conditions required to fix the shape of a C^ I shall in general 
 consider two curves to be the same when they are similar ; or, stating the same 
 thing analytically, two curves z=if(t) and x=.f^{t) will be considered the 
 same when they are so related that there exists between the points of the two 
 curves a one-to-one correspondence of the form z = mx -\- n, where m and n are 
 arbitrary complex constants. A correspondence or transformation of this kind 
 may be called a proportion ; it leaves the shapes of figures entirely unaltered, 
 merely changing their size and position in the plane. 
 
 In the method of discussion which I shall follow, it will be found convenient 
 to take the center of the curve as the origin and to make a^ equal to unity ; the 
 
ON THE PENTACAKDIOID. 3 
 
 equation in this form will be called the standard equation. The standard line 
 and map equations of the general (7^ are therefore : 
 
 x+ 5t+ lOat^ + 10bt^+ ^t^ + i/t^ = 0, (3) 
 
 x = — it—6at^—'ibt^ — t\ (4) 
 
 The equations in this form evidently depend upon and are entirely determined 
 by the single complex number a, which T shall therefore call the determining 
 coefficient of the curve. The equation of any C^ can be reduced to this form* 
 by means of a proportion ; for let the equation of the curve be given in its most 
 general form 
 
 z = Oo — "^pi^it — Gp.^xr,t^ — 4(p2/x2)<^ — {pi/xi)t^ ; 
 
 if in this we replace t by xfH (a change which leaves both the curve and the 
 
 reference system unaltered), and then divide by p^xf^ and transpose the constant 
 
 term, we have : 
 
 (z — ao)/pi4'^ = — 4< — 6pxt^ — 4{p/xy — t* 
 
 fi 
 
 where p^pg/pi and x=:x^lx\^. And now the equation is in the standard form. 
 
 This reduction is unique excepting as to the ambiguity introduced by the 
 process of taking the cube root of xj, and hence it can be made in three, and only 
 three, ways. We thus arrive at the following important theorem : 
 
 Two G^'s are similar when, and only when, the determining coefficient in the 
 standard equation of one curve differs from the determining coefficient of the other 
 curve at most by a cube root of unity. 
 
 Furthermore, since replacing a number (or a point which the number repre- 
 sents) by its conjugate is equivalent to a reflexion in the axis of reals, it follows 
 that if the determining coefficient of one G^ differs from the conjugate of the deter- 
 mining coefficient of another C" at most by a cube root of unity, the two curves are 
 inversely similar ; i. e., they are reflexions of each other in the axis of reals and 
 so bear to each other the relation of an object and its image in a plane mirror. 
 
 § 2. Construction of the Curve. 
 
 A construction for any C" depending on certain properties of its osculants 
 was described by Professor Morley in § 6 of the Memoir on Reflexive Geometry 
 
 •There 1b the one exceptional case of the cnrves for which the coeflBcient Oj In equation (2) la zero, but 
 we can consider this as a limiting case and include it In our standard equation by allowing the determiniDg 
 coefficient a to take the special value co ; and It should be borne in mind that \a\ is really the ratio 
 of |a,| to \a^\. 
 
4 ON THE PENTACARDIOID. 
 
 already referred to ; but there is another method, also mentioned by Professor 
 Morley,* which is more easily applied to the case of the C^ and I shall here set 
 forth a development of this second method. Refer to the standard equation of 
 the C^ and consider the two special curves corresponding respectively to the 
 values of the determining coefficient a = and a-=- (*:> . They are 
 
 Xi + 5<+5<* + j/i<«=0, (5) 
 
 a;3 + 10<'=+ 10<^ + y,<« = 0; (6) 
 
 with map equations 
 
 x, = — U-t\ (7) 
 
 x^^ — Qe—U^. (8) 
 
 These curves are two epicycloids easily constructed by simple mechanical 
 means ; the first is the three-cusped epicycloid traced out by a point on the cir- 
 cumference of a unit circle rolling about a circle of radius 3, whilst the second is 
 the one-cusped epicycloid traced out by a point on the circumference of a circle 
 of radius 4 rolling about a circle of radius 2. From these two curves all other 
 C^'s can be built up very simply. We must first notice that any curve of the 
 form 
 
 x.i= — Qat^—Aht^ (9) 
 
 is derived from (8) by multiplying x^ by px^, where a = pz = pe". For if in (9) we 
 replace t by xH, we have 
 
 aja 
 
 — QpxH^ — 4px7' = fx% . (10) 
 
 Now the effect of the factor px^ on the curve is merely to rotate it through the 
 angle 50 and to enlarge it in the ratio p : 1. Looking again at the standard 
 equation (4) of the general C^, we see that it is the sum of two simpler equations, 
 namely (7) and (9) ; therefore any C^ can be constructed by adding the vectors 
 that join the origin with corresponding points of two epicycloids properly placed. 
 By corresponding points are meant points on the two curves given by the same 
 value of t. Since the clinant of the tangent to all these curves at the point 
 given by the parameter t is — t^, it is evident that at corresponding points of 
 the two curves (7) and (9) the tangents are parallel, and owing to this fact we 
 can readily determine as many pairs of corresponding points as we wish. We 
 must, however, choose the proper starting points on the two curves, for there 
 
 ♦Orthocentrlc Properties of the Plane »4-Llne : Transactions of the American Mathematical Society, Vol. IV 
 (1903), pp. 7 and 8. 
 
a 
 
 a 
 
 ^ 
 
 (6) 
 
6 ON THE PENTACAEDIOID. 
 
 are five tangents to a G^ in any given direction, and according as we associate 
 one or another of these five points on the second curve with a selected point on 
 the first we obtain five different resulting C^'s. This ambiguity can be obviated 
 by choosing a definite value of the parameter t and determining the points given 
 by this value on the two curves by the ordinary process of plotting. Thus, putting 
 t = 1, say, we have aij = — 3 and a-g = — 6a — 46 ; these points are very easily 
 found as soon as we know the value of a, and taking this pair as our starting 
 points all further ambiguity is impossible. 
 
 In practice, it is much more convenient to take the mean of two points than 
 to take their sum, since to get the mean we need merely bisect the segment 
 joining the points; and for our purpose the points (xj -\- x^l2 are just as good as 
 the points (xj + a-g). Moreover, if we wish to alter the relative sizes of the two 
 fundamental curves, which is equivalent to altering the modulus of the deter- 
 mining coeflicient of the resulting curve, all we need do is to divide the stroke 
 joining corresponding points in a different ratio than 1:1; thus if we wish the 
 curve resulting from compounding (7) with the curve a; = — 6p<^ — 4p<^, we 
 divide the stroke joining corresponding points of (7) and (8) in the ratio p: 1. 
 Finally, when we wish to alter the amplitude of a, we merely rotate one of the 
 curves (7) or (9) through an angle equal to five times the desired change in the 
 amplitude. The combination of these two changes gives us every possible 
 variation in the value of the determining coefficient, and this method has been 
 found in practice to be a very simple and convenient one for obtaining all the 
 distinctive types of C^ The shifting of one or both the fundamental curves 
 without rotation does not affect the shape or size of the resulting curve and it is 
 more convenient to place the component curves some distance apart than to have 
 them concentric. 
 
 Figure II shows a number of C^'s constructed by this method,* and Figure I 
 illustrates clearly the carrying out of the method in two particular cases. The 
 two broken curves (a) and {d) are the fundamental curves in this particular case ; 
 their equations are (disregarding the constant term which has no effect on the 
 shape or effective position) : 
 
 xi = — 3^ — 2^ curve (a), 
 
 arg = + 4< + <* curve {d). 
 
 * The curves in Figure II are free-liaud reproductions, on a reduced scale, of larger figures constructed In 
 the same manner as Figure I. 
 
M S.^^/g-'e. 
 
 •fV 
 
 (il S, = U" 
 
 (c) S.'i'/v^c" 
 
 U) S, = 31>, it" 
 
 (e) S, = fiit'^ 
 
 (/) S.-.^/^c'e 
 
 b) s.^iU^ 
 
 (c-J S, = %U c'l^ 
 
 U) s, = :^'ixL^c> 
 
 -t" c g XX.Y e, E. 
 
 ') -S. - 6ci£.- 
 
 (7) 
 
8 
 
 ON THE PBNTACAEDIOID. 
 
 On each of these curves are marked thirty points such that the angle between 
 
 the tangents at any two successive points is 30 degrees ; the tangents at the 
 
 points 1, 7, 13, 19, 25 on either curve are all parallel to the 'tangent at the 
 
 point 1 on the other curve, and hence any one of those five points could be taken 
 
 as a corresponding point to the point 1 of the other curve ; the points marked 1 
 
 on the two curves were chosen as a corresponding pair. The strokes joining 
 
 these two points and the other corresponding points were divided in the ratio 
 
 1 : 2, and thus were obtained thirty points of the curve (6), of which the 
 
 equation is 
 
 3a; = 2a;i -\- x^ z= 4t — 6f — 'ifi + <*. 
 
 Next the strokes were divided in the ratio 2:1, giving the resulting curve (c), 
 of which the equation is 
 
 Sx = Xi + 2x2 = St—Sf— 2f+ 2tK 
 
 The values of the determining coefficient for the four curves, taken in the order 
 (a), (J), (c), {d), are respectively oo , — 1, — 1/4, 0. 
 
 § 3. The Singularities of the C^. 
 
 In order to determine completely the singularities, we need the ordinary 
 equation of the curve, namely, an algebraic relation between x and y independent 
 of <; we can best secure this by eliminating t between equation (2) and its con- 
 jugate. Performing the elimination by Sylvester's dialytic method, we arrive 
 at the desired equation in the following determinant form: 
 
 1 
 
 y 
 
 4b 
 
 4 
 
 6a 
 
 6b 
 
 4 
 
 4a 
 
 Since x and y appear only in alternate rows, and one in each column, the 
 developed equation will have as its highest term a;*?/*; hence a C" is of order eight. 
 Since the four highest powers of x, and also of i/, are absent, the curve will have 
 a four-fold point at infinity on each of the axes a; = and y^=0; i. e., at the 
 
 X 
 
 4 
 
 6a 
 
 46 
 
 1 
 
 4a 
 
 6b 
 
 4 
 
 
 
 X 
 
 4 
 
 6a 
 
 
 
 1 
 
 •4a 
 
 66 
 
 
 
 
 
 X 
 
 4 
 
 
 
 
 
 1 
 
 4a 
 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 y 
 
 
 
 
 
 46 
 
 1 
 
 
 
 4 
 
 y 
 
 
 
 6a 
 
 46 
 
 1 
 
 66 
 
 4 
 
 y 
 
 = 0. 
 
 (11) 
 
ON THE PENTACAEDIOID. 9 
 
   circular points, /and J. The symmetry of (11), with respect to x and y, shows 
 that the curve is symmetric in its relations to the two circular points. A four- 
 fold point being equivalent to six simple double points, these two singularities 
 are equivalent to 12 double points; and as a rational curve of order eight has 21 
 double points, there must be nine other double points, and in general these all 
 lie in the finite part of the plane. Now the condition for a cusp is the vanishing 
 of the derivative of x as to <, or : 
 
 1 + 3a^ + 3i<2 + <3 = o. (12) 
 
 As this is a self-conjugate equation in t, it has in general three turns as its roots; 
 and hence every C* has three cusps, and only three, of which two, however, may 
 be imaginary. Thus the nine double points comprise 3 cusps and 6 nodes. 
 We are now in a position to resolve the singularities at /and /; for if we put 
 d = number of nodes at each of the circular points, and Ic = number of cusps, 
 and note that the C^ is of class five, as is shown immediately by equation (1), 
 then Plucker's equation connecting the order and class of a curve gives us 
 
 5 = 56 — 2{2d + 6) — 3{2k + 3), or 2d + 3k = 15. (13) 
 
 Moreover, as we have seen, d + k = 6, and hence d = k = 3; so that each of 
 the four-fold points is equivalent to three nodes and three cusps. 
 
 With regard to the line singularities, it is to be noticed that the clinant, 
 being equal to — t^, has as its derivative — 5t*, and as this can not vanish for a 
 turn, the C^ has no inflections; and by using the appropriate Pliicker equation, 
 we find that there are six double tangents. This completes the enumeration of 
 the simple singularities of the curve. 
 
 By precisely analogous argument we can show that the general C" is of 
 class n and order 2(n — 1), that it has an (w — l)-fold point equivalent to n — 2 
 cusps and (n — 2)(n — 3)/2 nodes at each of the circular points, that it has 
 n — 2 cusps and (n — 2)(n — 3) nodes in the finite part of the plane, and finally 
 that it has (n — 1)(« — 2)/2 double tangents and no inflections. 
 
 It may of course happen that some of these singularities besides those at / 
 and J are imaginary ; and I would note that there are two views which may be 
 taken of such cases : The first is the projective view, which has been followed in 
 the above discussion and treats all the singularities, both real and imaginary, 
 as on the same footing; the second view is the one more proper to the present 
 method of analysis, which may be called metrical geometry, and which acknowl- 
 edges as properly belonging to the curve only the real nou-isolated singularities. 
 
10 ON THE PENTACAEDIOID. 
 
 In what follows, unless the contrary is stated, I shall confine myself to the second 
 
 view and deal only with the real non-isolated singularities, and one of my main 
 
 objects will be to determine how many such belong to any given curve and what 
 
 effect their presence may have on the form or shape of the curve. 
 
 Let us first consider the cusps. As we have seen, the cusp parameters are 
 
 given by the self- conjugate cubic (12). Since this is of odd degree, one of its 
 
 roots is necessarily a turn and hence every G^ has at least one cusp. The other 
 
 two roots of (12) may be either turns or a pair of inverse points; in the latter 
 
 case, the G^ has only one (real) cusp. If the three roots are turns, we have 
 
 three cusps, in general distinct; two roots of (12) may become equal for special 
 
 values of the determining coeflBcient a, and in this case two cusps come together; 
 
 the resulting singularity, as will be shown immediately, involves also a node, 
 
 and is a triple point with three coincident tangents and through which there 
 
 passes only a single branch of the curve. Such a point does not differ materially 
 
 in appearance from an ordinary point of the curve. For the proof it will be 
 
 found convenient to have the standard equation expressed in terms of the cusp 
 
 parameters. From equation (12) it is plain that if t^, t^, ta are the cusp 
 
 parameters, then 
 
 ti + tz + t3 = si = — 36, 
 
 hh + ¥3 + ¥1 = Sg = 3a,   (14) 
 
 Hence the standard equation may be expressed in the form 
 
 Sx = — 12t — 6s/ -f 4Sjf — 3<*. (15) 
 
 This equation shows incidentally that the shape of the curve is determined uniquely 
 as soon as the cusp parameters are given. 
 
 If two roots of (12) become equal, then t^ =^2 ^^^ ^3 = — 1/^! so that the 
 equation of the curve is, in this case, 
 
 Bx = — -[2t — 6{tl— 2lfiy + 4{2ti—l/^y—3t\ (16) 
 
 and the double cusp, given hy t = ti, is 
 
 3c = — 4ti—t\. (17) 
 
 A pencil of lines through this point is given by the equation 
 
 3c—3x — r{Sd—3y) = 0, (18) 
 
 T being the variable clinant and d the conjugate of c. If we substitute in this 
 equation the values of a; and c given by (16) and (17), we get an octavic in t the 
 
ON THE PENTACAEDIOID. 11 
 
 roots of which are the parameters of the eight points in which the line intersects 
 the C*; the result is 
 
 (<-0'(4/«? + ^ + 0-<iA-i/OW + iA+i/0 = o, (19) 
 
 and hence t = ti is a triple root for all values of r, so that the point is a triple 
 point. In like manner, by substituting the values of s^, Sg and x in the line 
 equation, namely, 
 
 3x+ 15t^ lOss^— 10s/ + 15<* + Syt^=0, (20) 
 
 we find the clinants of the tangents passing through this point to be given by 
 
 hence the three tangents at the triple point in question coincide. 
 
 It may further happen that the three roots of (12) are equal; in this case 
 the three cusps unite along with three nodes and form a quadruple point with 
 four coincident tangents.* Such a point does not differ in appearance from an 
 ordinary cusp. 
 
 The Nodes. The analytic condition for a node on a curve given by an 
 equation of the form x=/{t) is evidently [/(O — f{h)Vi^i — h) — ^- In the 
 case of the C^ we have then, from equation (15), 
 
 12 + 6*2(^1 + — 45j(<f + t,t, + 4) + 3(<f + t% + titl + <i) = 0. (21) 
 
 This, along with its conjugate, which is of degree six, namely 
 
 12tltl— QsMih + Q + WMti + tit, + 4) + 3(<f + 4t2 + tA + 4) = 0, (22) 
 
 if solved for t^ and ^ , would give us the eighteen parameters of the nine double 
 points; but this direct method of attack involves algebraical difficulties which 
 preclude any reasonable hope of success. I have therefore abandoned any direct 
 investigation of the nodes; an indirect method leading to a partial solution of 
 the problem will be described further on (in the following section). 
 
 With regard to the combination of simple double points into higher singu- 
 larities, we have the following limitation : Since the G^ is of class five, there 
 can not be any proper multiple points of order higher than two; i. e., no points 
 
 *The proof that a triple root of (12) does produce such a quadruple point on the C* Is entirely analogous 
 to that given in the case of a double root, and therefore need scarcely be given here. 
 
12 ON THE PENTACAEDIOID. 
 
 through which pass more than two distinct branches of the curve. Of improper 
 higher multiple points we have three kinds : 
 
 (a) A triple point through which there passes a single branch of the curve ; 
 this results from the combination of one node and two cusps ; 
 
 (b) A triple point through which there pass two branches of the curve ; 
 this results from the combination of two nodes and one cusp; 
 
 (c) A quadruple point through which there passes a single branch of the 
 curve. 
 
 The singularities (a) and (b) impose but one condition on the curve and hence 
 there is a single infinity of C^'s with each of these singularities ; (c) requires that 
 (12) be a perfect cube, so that there is only one form of C^ which has a quadruple 
 point. 
 
 It can happen that two nodes come together without further complications, 
 in which case we have a tacnode. This case leads to some results of special 
 interest. Equation (21) is the general condition which the parameters of a node 
 must satisfy ; for a tacnode they must satisfy the further condition <2 = e% 
 imposed upon them by the fact that the two branches of the curve are now 
 parallel. Combining this condition with (21), we have the necessary and 
 suflScient condition for a tacnode ; namely, 
 
 12 + 6s2(l+e")«— 4si(l+e"+e''")<2+3(l + e"+e^"+e^")<^=0, 72 = 1,2,3,4, (23)„ 
 or in somewhat more convenient form, 
 
 12+ 6S2( 1 + e^)t + 4si£'"( 1 + f")<2 — 3£*"f = 0. (24)„ 
 
 Whenever this is satisfied by some value of t which is a turn, the C^ will have 
 a tacnode; as the equation is not self-conjugate except for special values of Sj, 
 we must determine what values of «i will cause (24)„ to have a self-conjugate 
 factor. There are several methods for determining this condition on «i, but the 
 best is to eliminate Sg between this equation and its conjugate formed on the 
 supposition that t is a turn and then to solve the resulting equation for Sj in terms 
 of t; this gives us the map equation of a curve such that when s^ is any point 
 on it the corresponding G^ will have a tacnode. The value of s^ obtained in the 
 
 above way is 
 
 *, = 3(1 If + 2e''»«)/ 2(f'" + 6<"). (25)„ 
 
 If we replace t by e^"< and remember that l/(e^" + e'") = — (e" + e*"), this can 
 be put in the form 
 
 Si = -(3/2)(6" + 6*")(l/<^ + 20.   (26)„ 
 
  ON THE PBNTACARDIOID. 13 
 
 Giving n its four values, we get only two curves, since e" + e*" takes only the 
 two distinct values e + e^ and e^ + e' » each of these two curves is a three-cusped 
 hypocycloid, or, as it is sometimes called, a deltoid. When Sj is a point on either 
 of these curves, given say by the value t=zti, then (24)„ is satisfied identically 
 by the value t:=ti; and as ti is a turn, it follows that the corresponding C" 
 has a tacnode. 
 
 Finally, three nodes can combine into a double point of a special kind 
 known as an oscnode ; at such a point the two branches of the curve have three- 
 point contact, whereas at a tacnode they have only two-point contact; similarly 
 the tacnodal tangent is equivalent to two double tangents, the point of contact 
 counting as four points of intersection with the curve ; whilst the oscnodal 
 tangent is equivalent to three double tangents, and the point of contact counts 
 as six intersections with the curve. There are two G^'s possessing such a 
 singularity; the corresponding values of Sj will be derived in the next paragraph. 
 
 The Double Tangents. "We take the line equation in terms of the cusp 
 parameters ; namely, 
 
 3x+ 15t+ 10s/— 10s/ +15^* + Syt^ = 0. (20) 
 
 If we call ti and ^g the parameters of the points of contact of a double tangent, 
 these parameters must satisfy the two following relations : 
 
 15^1 + 10s/— 10s/ + 15t\ = 15^2 + 10s/- lOsi^ + 15<*, ] .^7) 
 
 q = <2 > J 
 
 with the condition ^i^^g. Combining these two relations, we get the four 
 following cubics whose roots give the twelve parameters of the six double 
 tangents : 
 
 3 + 2S2(1 + £")< + 2sj(e3» + e*")<2 — Se'^'f = ; n = 1, 2, 3, 4. (28)„ 
 
 It should be noted that the three roots of (28)^ are equal to the three roots of 
 (28)j, each multiplied by s; and the three roots of (28)3 are equal to those of 
 (28)2, each multiplied by e^ Thus the six double tangents are separated into 
 two sets of three which are distinguished from each other by the following 
 geometric property: Any tangent given by the equation (28)i is such that if we 
 pass along the curve in the positive direction from the first point of contact to 
 the second, we do not pass through any point at which the tangent is parallel 
 to the double tangent; whilst for any double tangent of the second set, given by 
 a root of (28)a, we always pass through one, and only one, point at which the 
 
14 ON THE PENTACARDIOID. 
 
 tangent is parallel to the double tangent. Since each of these equations is self- 
 conjugate and of odd degree, it follows that every C* has at least two double 
 tangents and, when there are only two, they belong to two different sets. 
 
 It should be noted that if we consider Sj as the variable and i as a parameter, 
 then (28)„ is the equation of a family of lines, or rather of two families of lines, 
 and the envelopes of these lines are precisely the two deltoids (25)„ already 
 found in connection with the tacnodes. That this is the case follows from the 
 geometric connection between the double tangents of a curve and the tacnode; 
 analytically it is shown very simply by the following form of the discriminant 
 of (28)„, which as usual gives the values which the variable (the variable here 
 being Sj) takes for all points on the envelope of the system. Calling t^, t^, t^ 
 the roots of (28),,, we have 
 
 <i + <2 + '3 = (2/3K(e*" + l); Ws = ^ "• (29) 
 
 Now if two roots become equal, we can put ti=.L, and therefore t^^=s"lt\; 
 so that the map equation of the discriminant is 
 
 2<i + eV<f = (2/3)si(f^" + l), • (30)„ 
 
 which being solved for s^ gives 
 
 s, = (3/2)(2<, + e"/<f)/(f*" + 1) ; (31)„ 
 
 and if we substitute e"< for <i, we have an equation identical with (26),„ and this 
 proves the fact in question. 
 
 For certain values of Sj, (28),, becomes a perfect cube and three double 
 tangents unite and have contact at one point only, which is, however, a double 
 point of the curve ; when this happens, we have a C^ with an oscnode. From 
 equations (29) we see that (28)„ is a cube only when if = c", which requires that 
 »i should have one of the two values (9/2)e"/7(«*" + 1)> ^= 1> 2. 
 
 Note (added June, 1908). For completeness we should add the following 
 compound singularities : 
 
 Three coincident double tangents giving a triple tangent; the necessary and 
 sufficient condition is 
 
 sj = — (3/2)[(l + e")< + l/<2](e2" + f*") ; n = 1, 2, 3 or 4. (32) 
 
 Four coincident double tangents; this singularity includes a tacnode and 
 may be called a tacnodal triple tangent. The condition is 
 
 s, = — (3/2)(e" + £*"), or Sj = — (3/2)(2 + e")(e^" + e*") ; w = 1, 2, 3 or 4. (33) 
 
::?•- 
 
 C4) 
 
 ^; 
 
 U) 
 
 (e-l 
 
 If} 
 
 Ul 
 
 U) 
 
 (^) 
 
 {^J 
 
 Fio. III. — Compound singularities of tlie C. 
 
 (15) 
 
16 ON THE PENTACARDIOID. 
 
 Six coincident double tangents giving a quadruple tangent. The condition is 
 
 «i = — 3/2. (34) 
 
 For convenience I append a table of all the compound singularities, including 
 those mentioned in the above note, and give references to the figures in which 
 occur the several singularities and the penultimate forms of the same. I should 
 remark that the curves in Figure III are schematic, but resemble quite closely 
 the actual curves. 
 
 Singularities. Condition on Si- Figures. 
 
 1. Triple point, 1 branch *i=2<— l/i^ HI, (<^) 
 
 = 2 cusps, 1 node 
 
 2. Triple point, 2 branches Sj = . ? . II, (i); III, (e) 
 
 = 2 nodes, 1 cusp 
 
 3. Quadruple point, 1 branch Sj = — 3 III, [i) and (/) 
 
 = 3 nodes, 3 cusps 
 
 4. Tacnode Si = — (3/2)(2< 
 
 = 2 nodes, 2 double tgs. + f7<2)(e''" + 6*") I, (d) 
 
 5. Oscnode «! = — (9/2)(e" + e*") HI, (n) and (o) 
 
 = 3 nodes, 3 double tgs. 
 
 6. Triple tangent Si = — (3/2)[(l + £")< 
 
 = 3 double tangents + 1 /t^lie^" + f*") III, (a) and (/) 
 
 7. Tacnodal triple tangent s, = — (3/2)(e'" + f'") HI, (b), (j) and (0 
 
 = 2 nodes, 4 double tgs. or — (3/2)(2 + e'')(e^"+e*») 
 
 8. Quadruple tangent Si = — 3/2 HI, (»«) 
 
 = 6 double tangents 
 
 . § 4. The Complete System. 
 
 Thus far I have considered the properties of the G^ in a somewhat isolated 
 manner; it is my purpose now to take up the sets of C^'s which are connected 
 together by the possession of certain special properties or certain common forms 
 of singularities, and I shall devote this section to what may be called the com- 
 plete system of C^'s. By the complete system I mean any collection of curves 
 including within itself all the double infinity of forms or shapes which the C^ 
 can have. In connection with this system of C^'a I shall consider certain singly 
 
ON THE PENTACARDIOID. 17 
 
 infinite sub-systems, and also certain interesting regions, loci and envelopes 
 which naturally present themselves in the course of the investigation and some 
 of which we have already come across in deriving the equations of condition 
 treated in the previous section. 
 
 The simplest analytical expression for this complete system is the standard 
 equation which we have been using above, namely 
 
 Sx = — 12t—6s2t^ + 4Sit^—3t\ (15) 
 
 or, in line form, 
 
 Sx + 15t + I0s2<2— 10s/ + 15<* + 3t/t^ — 0. (20) 
 
 When in either of these equations we allow the determining coefficient s^, or, 
 more conveniently for our purpose, Sj, to take the double infinity of values of 
 the complex number of the binary domain, we get all possible types of (7^ We 
 mp,y therefore consider Sj as representing a point of the plane, and two things 
 are to be noted : first, that though s, is subject to the condition that it must 
 always be the sum of three turns or of one turn and a pair of inverse points, 
 this condition does not impose any restriction on the value of s-i ; and secondly, 
 that owing to the relation between the determining coefEcient and the shape of 
 the curve established at the end of § 1, we may without any loss of generality 
 restrict Sj to any third portion of the plane bounded by a pair of straight lines 
 or rays through the origin. We shall make use of this restriction on Sj later on. 
 I shall begin this investigation with the cusps. The first fact that strikes 
 our attention is that though we have a double infinity of curves yet they have 
 but a single infinity of cusp tangents ; for on combining the condition for cusps, 
 which is simply 
 
 l-\-8zt — Sit^ + i^=0, (35) 
 
 with the line equation (20) of the system, both Sj and s^ are eliminated together, 
 giving as the envelope of the cusp tangents the curve 
 
 Sx+ 5t+ 5t^ + Syt^=0.. (36) 
 
 This is a three-cusped epicycloid concentric with the whole system (20) and has 
 
 for its map equation 
 
 Sx = — 4t — 1\ (37) 
 
 Each line of this curve, therefore, must be a cusp tangent for a whole infinity 
 of C^'s of the system. But it must be noted that not any three tangents of (36) 
 can be chosen as cusp tangents of a C^ of the system ; for the cusp parameters 
 
18 ON THE PENTACARDIOID. 
 
 of any curve of the system are subject to the involution determined by equation 
 (35), namely Sg -]- 1 = 0, and therefore when two of the tangents are chosen, the 
 third is uniquely determined. 
 
 The epicycloid (36) has some further interesting connections with the cusps, 
 which I shall here set forth. For this purpose I shall consider the behavior of 
 the three cusps when we hold one of the cusp parameters fixed. The cusps of 
 the complete system are all included in the formula 
 
 3c„ = - 9<„ — Ss^tl + Sitl, n = l,2, 3, (38) 
 
 where <„ is a root of (35). To study the behavior of the cusps, it is best to 
 
 replace Sj and s^ by their values in terms of <„ and to simplify the equations by 
 
 means of the involution Sg + 1 = 0. We thus obtain, as the equations of the 
 
 three cusps, 
 
 3ci = — 6^1 + 2tl/t^ — 2t% + t{, ' 
 
 Sc^ = — 6<2 + 2tl/t, — 2tlt, + 4,   (39) 
 
 3c3 = — 6^3 + 2tyti — 2tlt, + tl . 
 
 This of course is not the only form in which the equations can be put ; and by 
 replacing one of the parameters by its value derived from the involution 
 Sg + 1 = 0, we get another form which will be found useful to us, namely 
 
 3c3 = 6/t,t, + 2/tltl + 2lt\tl + \lt\tl, (40) 
 
 and two similar equations obtained by cyclically interchanging the subscripts. 
 
 Now let us hold one of the parameters, t-^ say, fixed and allow the other two 
 to vary, replacing them by t. Under these conditions the three cusps will have 
 to move along definite curves, since their positions depend on the single variable t. 
 The cusp Ci is by this means separated, as it were, from the other two, and these 
 latter lose their identity and may be treated as being a pair interchangeable at 
 will. The cusp Cj moves along one curve, namely 
 
 3ci = — 6<i + 2t\jt— 2t\t -\-t\, (41) 
 
 and the other two move along a second curve given by 
 
 3c = — 6< + 2tyti — 2t\ + <* (42) 
 
 or the equivalent equation 
 
 3c = 6/<i^ + 2lt\e +" 2lt\t^ + 1 lt\tK (43) 
 
 Equation (41), the locus of Cj, represents a straight line, or rather a seg- 
 ment of a line described twice over; the cusps or extremities of this segment 
 
ON THE PENTACARDIOID. 19 
 
 are given by the values of t determined by the equation dci/dt= 0; i. e., by 
 t% +1 = 0. The ends of the segment are therefore 
 
 3^ = — 6<i ± 4i<P + t\. (44) 
 
 The length of this segment, or the distance between the two points given by 
 (44), is 8 ; and it should be noted that this is independent of the value of t^. 
 
 It is quite clear, apart entirely from the equation (41), that the cusp Cj must 
 move along a straight line when t^ is fixed, since this means that the cusp tangent 
 on which Cj lies is a definite fixed tangent of the epicycloid (36). The above 
 results give us this further information, that the cusp does not travel along the 
 whole line but is restricted to a finite portion of length 8, no matter what 
 tangent of (36) we may choose as the cusp tangent. 
 
 The other two cusps move along the curve (42) or (43). This curve is an 
 octavic with two cusps given by the same values of t which determine the cusps 
 of the segment (41); namely, by t=. dzijtl'^; the cusps of the curve (42) are 
 therefore 
 
 ZG=±4iltY^-l/tl. (45) 
 
 Now suppose we give another value to ti; we shall evidently get two new curves, 
 another segment for Cj and another octavic for Cj and Cj ; and if we suppose t^ to 
 vary continuously, we shall get two families of curves, and the cusps of the two 
 families will lie on certain curves; namely, the curves obtained by letting t^ be a 
 variable in equations (44) and (45); but (45) is merely the equation of the epi- 
 cycloid (36) in slightly different form, as is clear when we make the substitution 
 t = i/ty^; hence this epicycloid is not only the envelope of the segments (41) or 
 cusp tangents, but also the locus of cusps of the octavics (42). In order to see 
 what relation this epicycloid has to the singularities of the system of C^'s, we 
 must look a little more closely at the values of t which give the cusps of (42). 
 As we saw above, these values are the roots of the equation t^f + 1 z=. 0, where 
 t stands for t^ or tg ; but tjt^ts +1 = always, and therefore, whenever Cg or Cg is 
 at a cusp of (42), tz = t^ and the two cusps coincide, so that we have a triple 
 point of the kind described on page 11. Thus we see that the epicycloid (36) is 
 the locus of all such triple points occurring in the complete system of C"s. This 
 fact can be proved more directly by putting t = tj^ in equation (42), for then Cj 
 coincides with either c^ or Cg, and at the same time the equation becomes iden- 
 tical with (36). From equation (40) we see that when q and Cg coincide, Cg is on 
 the curve 
 
 3x = 6<« + 4<» + fi, (46) 
 
20 ON THE PENTACARDIOID. 
 
 obtained by putting ^j =: ^g = Ijt in (40) ; and this is the same curve as (44), since 
 the two equations become identical when we make the substitution f ^ — <i. 
 
 It is an interesting fact that the complete envelope of the family of octavics 
 (42) consists of both the curves (37) and (46). The envelope can be obtained by 
 the ordinary process ; but this is much simplified by noting that equation (43) is 
 symmetric in t and ti, and when this is the case the envelope is given, at least 
 partially, by merely putting <= <i; This gives the curve (46). In order to get 
 the other part of the envelope, we must employ the usual process and apply it 
 to equation (42), when we obtain equation (36). It should be noted that if we 
 use only one form of the equation, we get only a part of the envelope, and to 
 get the complete envelope we must use both (42) and (43); this is due to the 
 fact that the epicycloid (36) is not only a part of the envelope but also the cusp 
 locus, and unless special precautions are taken the values of the parameter 
 giving the cusps factor bodily out of the equation of condition for the envelope. 
 The curve (46), besides being the partial envelope of the octavics, is also the locus 
 of the ends of the segments along which the cusps of the G^'& travel, and 
 therefore bounds the region of the plane within which all the cusps of the com- 
 plete system lie. We see, as a corollary of the theorem of § 1, that the curve 
 (46) must have triple symmetry ; this is also proved independently by merely 
 substituting i^t for t in (46) and noticing that we then get Sax in place of 3a;. 
 
 The rest of this section will be devoted to an investigation of the reality of 
 the singularities of the C^, and I shall set forth somewhat fully a graphical 
 method for the determination of this point arising from the equations of condition 
 expressed in § 3. The problem then is this: Given the equation of a G^, how 
 many real singularities has it? I shall take first the cusps, then the double 
 tangents, and lastly the nodes. Let then a definite curve be given, and I shall 
 suppose that its equation has been reduced to the standard form (15) or (20) 
 given above. The cusp condition is then the simple expression 
 
 1 + S3< — «/ + !!=' = 0, (35) 
 
 and the reality, coincidence and so forth of the cusps depend on the reality, 
 equality and other relations of the roots of this equation, and these relations 
 depend entirely on the value of the coefficient Sj. As in a previous case (p. 14), 
 so here also we may consider (35) as the equation of a family of straight lines, 
 Sj being the variable and t the parameter of the family ; (35) is therefore the 
 line equation of a curve, and the discriminant of (35) considered as a cubic in t 
 
ON THE PENTACAEDIOID. 21 
 
 will be the equation in map form of the point curve, or envelope of the family 
 
 of lines. Instead of forming the discriminant by the ordinary methods leading 
 
 to an equation involving only Sj and s^ as the variables, we follow the far simpler 
 
 method of expressing the discriminant in terms of Sj and t; and thus it is that 
 
 we get the equation in map form. If it were desirable, we could eliminate t 
 
 between this resulting equation and its conjugate and obtain the ordinary 
 
 equation of the curve. Since Sj = ^j -f ^3 -f ^3 and t^t^fs = — 1, the discriminant 
 
 is simply 
 
 si = 2t—l/t\ (47) 
 
 and this represents a three-cusped hypocycloid or deltoid. Now the deltoid is 
 a curve which has been carefully studied and whose properties are well known,* 
 and so it may be used with profit in the present investigation. Since for a fixed 
 value of Sj the three roots of (35) are the parameters of the cusps of the corre- 
 sponding C^, and at the same time the parameters of the tangents from the 
 point Si to the deltoid (47), we see at once that the C^ determined by any 
 particular value of Sj has as many real and distinct cusps as there are real and 
 distinct tangents from the point Sj to the deltoid. Hence when s^ is within the 
 deltoid, the C^ has three real and distinct cusps; when it is outside the deltoid, 
 the G^ has only one real cusp ; and when Sj is on the deltoid, the C^ has two 
 coincident cusps and therefore a triple point. There are three singular points 
 on the deltoid, namely, the three cusps, for which (35) becomes a perfect cube ; 
 and hence when Sj is at one of these points, the three cusps of the C^ must 
 coalesce, and therefore the curve will have a quadruple point. Since the three 
 cusps of the deltoid are equispaced about the center, which in this case is the 
 origin of coordinates, the three special values of Sj differ only by a cube root of 
 unity, and hence therie is only one C'^ having three coincident cusps. Thus the 
 position of Sj in reference to the deltoid (47) tells graphically, and at a single 
 glance, the story of the cusps of the corresponding C^. 
 
 Next let us examine the double tangents. We have already seen (p. 14), 
 that the discriminant of the double tangents consists of two other deltoids; 
 namely, 
 
 s, = - (3/2)(2< + e-ie){^^' + e'"). (3l)„ 
 
 * Of. J. Steiner: Ueber eine Kurve Dritter Klasse, Crelle's Journal, Vol. LIII (1857), pp. 331 ff., where the 
 curve is studied by the methods of synthetic geometry. For a treatment of the curve along the lines of the 
 analysis employed in the present paper, cf. F. Morley : Orthocentric Properties of the Plane n-Line, Transactions 
 of the American Mathematical Society, Vol. IV (1903), pp. 1 ff. ; »nd H. A. Converse : On a System of IlypocycloidB 
 of Class 3, Annals of Mathematics ^ Series 8, Vol. V (1904) 
 
22 
 
 ON THE PENTACAEDIOID. 
 
 The position of Sj in relation to these two deltoids tells the story of the double 
 tangents in the same way as its relation to (47) told the story of the cusps. 
 When si is on either of the deltoids, the C^ has a tacnode; when s^ is on both 
 deltoids, the C^ will have two tacnodes; and when it is at a cusp of either 
 deltoid, the C^ will have three coincident double tangents with a single pair of 
 contacts, and this singularity is the oscnode. 
 
 Before proceeding to the nodes, it will be convenient to examine the relative 
 size and position of the three deltoids just considered. For this purpose we will 
 compare the three equations (47), (26)i and (26)2. I shall call the discriminant 
 
 Fig. IV. — The discriminant deltoids of the complete system of C's. 
 
 of the cusps Z?o and the discriminants of the double tangents or nodes D^ and D^. 
 Since there is no constant term in the three equations, it follows that the three 
 deltoids are concentric; and as the coefficients are all real, it follows that the 
 axis of reals is an axis of symmetry for each of the deltoids. For the further 
 determination of the curves, we need merely determine the position of one of the 
 cusps of each curve. The cusps on the axis of reals are at the following points; 
 On Do at —3, on D^ at 9cos(27t/5) = — 2.78 +, on D2 at 9cos(47t/5) = 7.28 +. 
 Thus the cusps of Dq and D^ are pointed in the same direction, whilst those of D^ 
 are pointed in the opposite direction ; hence Dq and D^ do not intersect each 
 other at all, whilst Dg intersects both Dq and D^ in six points each. The three 
 curves are therefore as shown in Figure IV. 
 
ON THE PENTACARDIOID. 23 
 
 These deltoids divide the plane into a certain number of regions, and the 
 number of real singularities possessed by any given C^ depends on the region 
 in which the coefficient Sj of the given C^ lies. We need not consider all the 
 regions, since, owing to the theorem deduced at the end of § 1, the shape and 
 therefore all the singularities of the C^ are the same for any three values of s, 
 which are equispaced about the origin; and if two values of Sj are reflexions of 
 each other in any one of the three axes of symmetry of the deltoids, then the 
 two corresponding C^'s are images of each other and therefore have exactly the 
 same singularities. Hence, in considering the nature of the singularities, we can 
 
 Fio. v. —The sub-regions of s,. 
 
 restrict s^ to any portion of the plane bounded by two rays from the origin 
 making an angle of 60 degrees with each other. For convenience, we choose 
 for the limiting rays the positive half of the axis of reals and the ray through 
 the point — w^, both limits being included in the region of Sj . This region may 
 be called the essential region of s^. And now, by means of the three deltoids, 
 this region is divided into six sub-regions which will be designated by the 
 
 symbols R^, , R^. These regions are shown in Figure V, in which the 
 
 relative dimensions of the three deltoids have been slightly altered for the sake 
 of clearness. R^ is infinite in extent and includes all that portion of the essential 
 region of Sj which lies entirely outside of all the deltoids. 
 
 By invoking the principle of continuity and arguing from what we may 
 know about the C* for a particular value of «i, we can by means of these regions 
 determine almost everything about the singularities; the only difficulty arises in 
 
24 ON THE PENTACARDIOID. 
 
 trying to determine the actual number of nodes. By enumerating the number 
 of tangents that can be drawn to the deltoids from a point in any one of the six 
 regions, we know at once how many cusps and double tangents belong to the 
 corresponding C". With regard to the nodes, we can not say how many there 
 are for each position of S], but we can tell whether the G^ has an odd or an 
 even number. For the only way in which the C^ can acquire or lose nodes is 
 by passing through some intermediate complex singularity involving one or more 
 nodes; these singularities are the following:* 
 
 1st, Triple point with a single branch ; 2nd, Quadruple point with one 
 branch ; 3rd, Tacnode ; 4th, Oscnode; and 5th, Triple point with two branches. 
 In the first case the C^ loses or gains one node ; in the second, three nodes; and 
 in the last three cases, two nodes. Hence we may state generally that when 
 the C^ acquires a node for values of Sj other than those given by Z>o, it acquires 
 a pair of nodes. Now when Sj ^ 0, we have the three-cusped epicycloid, which 
 we know independently has no nodes ; therefore when «i is within Dq the C^ has 
 an even number of nodes, and when Sj is without Dq the G^ has an odd number 
 of nodes. We can now make out a table of singularities for the various regions, 
 the number of simple singularities possessed by a given C^ being placed opposite 
 the region in which the coefiGcient Sj of the given C^ happens to lie. 
 
 Region. 
 
 Cusps. 
 
 Db. Tgs. 
 
 Nodes 
 
 R,   
 
 3 
 
 6 
 
 Even 
 
 R, 
 
 3 
 
 4 
 
 Even 
 
 Rs 
 
 3 
 
 4 
 
 Even 
 
 R, 
 
 3 
 
 2 
 
 Even 
 
 A 
 
 1 
 
 4 
 
 Odd 
 
 i?« 
 
 1 
 
 2 
 
 Odd 
 
 This table of regions is incomplete in so far as the number of nodes is con- 
 cerned and must remain so until the equation of condition for the singularity 
 consisting of two nodes and a cusp is found. I shall here leave the subject, 
 with the hope that it may be completed at a later date. 
 
 Johns Hopkins Univkbsitt. March, 1908. 
 
 • Cf. Table of singularities, p. 16. The completeness of this table, upon which rests the validity of the 
 argument here used, has not as yet been rigidly proved; but careful investigation makes it almost certain that 
 there are no other compound singularities occurring on any &'. 
 
BIOGRAPHICAL NOTE. 
 
 Edward C. Phillips was born in Germantown. Pennsylvania, on November 
 4, 1877. His early education was secured in the Parochial Schools, and he made 
 his collegiate studies at the College of St. Francis Xavier, New York City, 
 graduating from that institution with the degree of Bachelor of Arts in 1898. 
 He then entered the Novitiate of the Society of Jesus at Frederick, Maryland. 
 PVom 1901 to 1904 he was at Woodstock College, Maryland, engaged chiefly in 
 graduate studies in Philosophy. In October, 1904, he came to the Johns Hop- 
 kins University and entered the department of Mathematics as a graduate 
 student. Since then, with the exception of the year 1906-7, he has been 
 following courses of Mathematics, Physics and Physical Chemistry. 
 
 March, 1908. 
 
BERKELEY ^'^^^' 
 
 ^HIS BOOK IS due"^ ^^_ ^^ 
 ao «-^ no. rl^^=^ BeS^^^«^ bate 
 
 20m-ll,'20 
 

 
 
 
 I- 
 
 ^.