LIBRARY 
 
 OF THE 
 
 University of California. 
 
 (ryvvv<>. \'TJrt:Jh,Ayy%<i i^'llf^^wor... 
 
 Class 
 
CERTAIN PARTIAL DIFFERENTIAL EQUATIONS 
 
 CONNECTED 
 
 WITH THE THEORY OF SURFACES 
 
 DISSERTATION 
 
 SUBMITTED TO THE BOARD OF UNIVERSITY STUDIES OF THE JOHNS HOPKINS UNIVERSITY 
 FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 
 
 BY 
 
 NATHAN ALLEN PATTILLO 
 
 1897 
 
 Of THI 
 
 VNIYER8ITY 
 
 C^e Motb (0afttmore (press 
 
 THE FRIEDENWALD COMPANY 
 BALTIMORE, MD., U. S. A. 
 
^5 
 
«x 
 
 INTRODUCTION. 
 
 In two notes in the Coraptes Rendus of Oct. 26, 1896, and Nov. 16, 1896, 
 and in the American Journal of Mathematics of Jan., 1897, Professor Craig has 
 considered certain partial differential equations connected with the theory of 
 surfaces. 
 
 In this paper the work done there has been continued. The first section is 
 introductory, and gives a brief resume of Laplace's method for the integration of 
 linear partial differential equations of the second order. 
 
 In the next two sections Professor Craig's equations {Eoi) and (£"02) 
 are deduced together with their equivalent equations and the corresponding 
 Laplace's series, and the general formulae for calculating their invariants are 
 derived. The form of substitution required to pass from an equation of one 
 series to one of the other series has also been found. 
 
 In the fourth section it is shown that by making the two equations {Eoi) 
 and {Eiiz) identical the lines of curvature form an isothermal system and the 
 reciprocals of the two principal radii of curvature satisfy adjoint equations. 
 
 The next two sections are taken up with the calculation of the coefficients 
 of the general transformed equations and making the general equations equal. 
 Here the linear element reduces to the isothermal form. 
 
 In the seventh section the hypothesis 6oi = ao2 = 0, has been made. Here 
 the equations (^01) and {E02) reduce to integrable forms and the lines of 
 curvature form an isothermal system. 
 
 The following section has been devoted to the consideration of the 
 hypothesis : 
 
 «oi = ^02 ) 
 
 <^02 ^^ f^Ol' 
 
 Then the conditions have been found that must exist in order that the 
 reciprocals of the principal radii of curvature of parallel surfaces may satisfy the 
 same partial differential equations as those of the original surface. 
 
 The forms to which the equations {Eoi) and (Eoz) reduce for the ellipsoid 
 and the cyclide of Dupin are then given and it is seen that they may be readily 
 integrated by the method considered by Poisson. 
 
 I desire to make this acknowledgment of gratitude to Professor Craig, who 
 suggested to me the subject of this dissertation and to whom I am indebted for 
 valuable assistance. 
 
 4 A iCi^a 
 
Digitized by the Internet Archive 
 
 in 2008 with funding from 
 
 Microsoft Corporation 
 
 http://www.archive.org/details/certainpartialdiOOpattrich 
 
CERTAIN PARTIAL DIFFERENTIAL EQUATIONS CONNECTED WITH 
 THE THEORY OF SURFACES. 
 
 1. Consider the linear partial differential equation 
 
 |!f_ + a|i' + 6|f + 0i. = 0, 
 
 (1) 
 
 where a, b, c are any functions whatever of u and v ; it can be put in either of 
 the two following forms : 
 
 du 
 
 Introduce the two functions h and k defined as follows : 
 
 (2) 
 
 (3) 
 
 A = ^ — \- ab — C) 
 du 
 
 k = ~4-ab — c . 
 dv 
 
 If h is zero, the first of equations (2) can be written 
 
 Therefore, the complete determination of all the integrals of equation (1) is 
 reduced to the successive integration of two equations of the first order 
 
 
 (4) 
 
 integration which requires only quadratures. 
 
 In like manner, if k is zero, the second of equations (2) can be put in 
 the form 
 
 or TH£ X 
 
 UNiVERSITY i 
 
6 Certain Partial Differential Equations 
 
 and we can replace equation (1) by the two equations 
 
 1^ +bf =<p_, 
 
 (5) 
 
 Suppose now that h and k are not zero. We shall show that these functions 
 enjoy properties of invariance, which have a great importance in the study of 
 the proposed equation. Consider successively the substitutions 
 
 <p = If' ; 
 
 U=zf{u'), VZ=(/f{l/)', 
 
 Make at first 
 
 X being any function of u and v. The equation in <p will become 
 
 (6) 
 
 (7) 
 
 If we calculate the new values of h and ky we find that these functions have 
 not changed. 
 
 Make now the substitution 
 
 u 
 
 =f{u'), v = ip{y^. 
 
 Equation (1) will become 
 
 ^li- + «i*' w aS + v {»') 1^ + «/ M (^ (»-) f = 
 
 The new values A', ^ of A and fc will be 
 
 Finally the substitution 
 
 h' = hf{u').p'{v/),i 
 k' = kf{u')iP'{i/).S 
 
 (8) 
 
Connected with the Theory of Surfaces. 7 
 
 will exchange the values of h and h. All these properties justify the name 
 of invariants which we shall give to h and h. 
 
 Suppose that the invariants h and h are not zero. Make the substitution 
 given by the equation 
 
 ^i = |f + «^. (9) 
 
 Equation (1) can now be written 
 
 |g + % = %. (10) 
 
 If we eliminate f between these two equations we get 
 
 where 
 
 |¥l + a,|^^ + 6,|^-^ + c,^, = 0, (11) 
 
 dlosh j^ 7 „ 3a , 86 ,3 log A, ,^^ 
 
 dv du dv dv 
 
 The value of the invariants will be 
 
 K=2h — k ^J|^ ,1 ^^3^ 
 
 k, = h. J 
 
 They are expressed only as functions of the invariants of the given equation. 
 Make also the substitution 
 
 We shall have 
 
 ^_, = || + 6^. (14) 
 
 ^ + «f'-i = %, (15) 
 
 and we get the following equation in ^_i 
 
 &^ + °-.%-+*-.%- + <'-*-. = 0- (16) 
 
 In this the coefficients have the values 
 
 «_, = «, 6_, = 6-^^^ c,=:c-i^, + ^-a^\ (17) 
 
 du ' dv du du 
 
8 Cebtain Partial Differential Equations 
 
 The values of the invariants will be 
 
 k^i=2k — h 
 
 logk^ > 
 hdv ' J 
 
 (18) 
 
 Thus we deduce from the proposed equation (E) two new equations (Ei) and 
 {E_i). We can apply the same method to these two equations, but we will 
 not get two new equations for each of them. The first of the two substitutions 
 applied to the equation (^_i) will give us the proposed equation in which 
 
 ^ will be replaced by -^ . The second substitution, applied to (Ei), will bring 
 
 us back to the proposed equation in which f will be replaced by -^ . 
 
 If then we regard as equivalent two equations which reduce to each othe 
 by changing <f into X<p and which have, therefore, the same invariants, we see 
 that the substitutions of Laplace applied successively will give only one linear 
 series of equations 
 
 ....,(J5;_,),(^_i),(^), (^0,(^2),.... 
 
 with positive and negative indices. 
 
 The invariants of equations {Ef) are deduced from each other by the repeated 
 application of equations (13) and (18). We find thus 
 
 Solved with respect to hi and ki, these formulas give 
 
 (19) 
 
 ki :=: 2«<4. 1 — hi^^ 
 
 aMogVhi 
 
 dudv 
 
 (20) 
 
 We can give to i all integer values, positive or negative. Instead of considering 
 two series of quantities hi and ki we could introduce only the quantities A< . We 
 shall then have the series 
 
 . . . . > ^_2> "■— 1> h f hi f hiy . . . , f 
 
Connected with the Theory op Surfaces. 
 deducing one from the other by the formula 
 
 A 4- A —2h— 3'MA< 
 
 (21) 
 
 The invariants of the equation (Ei) will be hi and hi_i. By the linear com- 
 binations of equations (21) we can obtain the relation 
 
 hi^, = hi+h-k- ^'^''^l\" • '^^ . 
 ^ dudv 
 
 (22) 
 
 2. Let u and v denote the parameters of the lines of curvature of a surface, 
 Pi and />2 the principal radii of curvature of the surface at the point (w , r) , pi 
 being the principal radius corresponding to the line v=z const, (the w-line) and 
 P2 corresponding to m= const, (the «-line) . Let i?i, R^ denote the radii of geo- 
 desic curvature of the lines u =. const, and v = const, respectively. 
 We have now 
 
 _E _G 
 
 1^ 
 
 R^ 
 
 1 
 
 — 1 dE 
 
 2E^G dv ' 
 — 1 dG 
 
 El ~ 2Ga^E du 
 
 (23) 
 
 We have also the following equations connecting all of the preceding 
 quantities 
 
 p{ dv R2 \pi Pz/ 
 
 ^2_VEf^_l_\_Q 
 du R^ \p^ p^J 
 
 1 dp 
 
 P2 
 
 (24) 
 
 Liouville's formula for the measure of curvature can be written 
 
 ^/EG_ d sfGt d s/E 
 
 piPi du Ri ' dv R2 
 
 (25) 
 
 Substituting the values of pi and p^ in equations (23), they can be reduced 
 to the form 
 
 d \/E a/Ea/G' 
 
 dv pi^ 
 d s/G 
 
 R2 p2 
 
 VG\/E 
 
 du p2 Ri pi 
 
 (26) 
 
10 
 
 Certain Partial Differential Equations 
 
 Differentiating the first of equations (24) for u and the second for v and 
 
 eliminating — and — respectively, we obtain the equations 
 Pi Pi 
 
 d' I VG d I 
 
 dudvpi B-i dupi \du ^^ R2 "^ Jii J dv pi 
 dudv (h \dv ^ El * R2 Jdu pz Ri dvp2 
 
 That is, - and -— are particular integrals of the differential equations 
 Pi Pi 
 
 / 7^ > _ 3^f 01 ^G 8f 01 / a 1 ^/G . VE\ 9f 01 _ 
 
 (27) 
 
 (28) 
 
 The differential equation satisfied by the coordinates {x, y, z) of the point 
 (u, v) of the surface may now be written 
 
 d^<Po ■ VG d<po ^/Edfii _ Q 
 dudv* B2 du ' Ri dv 
 
 From equations (26) we derive in the same way as above the following 
 
 dudv du ^R, dv R,R.,^''~ ' 
 
 .2^o2__a 1 y'^afo2_v^^ --0. 
 
 dudv dv ^ R, du R,R^ ^"^ 
 
 (29) 
 
 (30) 
 
 These equations have ^-— and — — respectively as particular integrals. 
 
 pi Pz 
 
 They are equivalent to equations (28) ; that is, they have the same invariants. 
 
 3. We shall now consider Laplace's series of equations corresponding to 
 (£"01) and {E^)y 
 
 . . . . , (J57_ii) , . . . . (J5/_ii) , (£'01) , (J?ii) ) (-^21) , • . • . (-Eii) , .... 1 .gjx 
 . . . ., (^_«),. . . .(^_x2), (^02), (^12), (^22), . . . . [E^), . . . ./ 
 
 Using equations (3), we find the invariants of (j^i) are 
 
 ^/EG . __ 
 
 dudv ^ ^2 "^ R1R2 ' 
 
 (32) 
 
Connected with the Theory of Sukpaces. 
 those of {Eo'i) are 
 
 The application of formula (22) will give the general invariants, 
 we find for all positive values of i 
 
 hn = h,_, , + ^^^ log ^5 - 3^^ log KA^ . . . . ^-. . 
 ^e = ^i-i, 2 — ci o log -^ KiKz' ' • -hi-uz' 
 
 OUQV Ki 
 
 K\ — - "•<— 1,1) \ 
 
 f^t2 =^ "i— 1,2* f 
 
 For negative values of i we have 
 
 A_« =A_,,_,,.,-^|^-log^^A_n/^_,,. . . .A-a-i),i, 
 
 ga a/ 6r 9^ 
 
 A_« =/l_„_,>.2 + g,^3-^ log -]R- -3^-3- log /^02A-12. . . ./^-(,-l).2 
 
 = ^-«-l).2-3^l0g-^^/l_12/l_22. . . ./l_U-l,.2. 
 
 11 
 
 (33) 
 Thus 
 
 (34) 
 
 (35) 
 
 k. 
 
 -a — "- «+i),i) \ 
 
 -C = "'-(i+l),2' J 
 
 From the first of equations (34) it may be seen that 
 
 (36) 
 
 (37) 
 
 Therefore, 
 
 , _s/EG d' 1 \^G 
 
 "•11 ^^^ ''■02 ) \ 
 
 "'ll =^ "01 ^^ n^02 • J 
 
 Consider the general invariants. Let us suppose 
 
 ""i — 1,1 ^^ "■< — 2,2 J 
 then it may readily be seen that 
 
 Hi "i-l,2» \ 
 
 (38) 
 
 (39) 
 (40) 
 
12 
 
 Ceetain Partial Differential Equations 
 
 But we have found equation (39) to be true for i=z2; 
 all values of i. Therefore, we have the same invariants in 
 equations (31), each invariant in the first series being equal 
 invariant in the second series. Then the equations {En) 
 equivalent. Hence, the first may be reduced to the second 
 function (pn by ^fi-1,2 • We can determine ^ by identifying 
 the two equations. By repeated application of equations 
 general coefficient «= 
 
 __ d log h Ai ' • • -^i-i.! 
 
 then it holds for 
 the two series of 
 to the preceding 
 and {Ei_i^2) are 
 by replacing the 
 the coefficients in 
 (12) we find the 
 
 a — 
 
 dv 
 
 — n 31og/to2 A<_2.a _^ d log/tn. . . .^<-l.l 
 
 Oj-1.2 — 002 %; —aoz — ^ , 
 
 Oj_l,2 ^ O02 • J 
 
 (41) 
 
 (42) 
 
 The formulas of identification which we may take from equation (7) 
 will give 
 
 j^ 3 log X 
 
 Oi_i, 2 — «.i i ^^ f 
 
 fe<-l,2=^*l + 
 
 d log X 
 
 (43) 
 
 Replacing o,_i,2 and a^ in the first equation by their values taken from 
 (41), we get the equation 
 
 d\osX_d.VG 
 
 After integration it may be placed in the form 
 
 log>l = log^ + logCr. 
 
 (44) 
 
 By substituting the values of 6<_i,2 and 6,1 taken from (42) the second of 
 equations (43) may be reduced to the form 
 
 Hence, 
 
 log ;= log -^ 4- log V. 
 
 (45) 
 
Connected with the Theory op Surfaces. 13 
 
 By comparing (44) and (45) it may be seen that U and V must be 
 constants. Therefore, neglecting a constant factor, which would disappear after 
 substitution, we may take 
 
 
 Ro 
 
 \/G 
 Hence, if in (E^) the function f^ be replaced by — p- ^,_i 2,weshall find (JS',_i,2)' 
 
 \/E 
 In the same way it may be shown that replacing ^_<2 in {E_i2) by ^5- 
 
 yj-^-i),! will give the equation (^_(j_i),i) , 
 
 4. We shall now consider the result of making the equations {Eoi) and (£"02) 
 identical. By examining the coefficients it is evident that we have the condition 
 
 l-^^=^' 
 |-^^='>- 
 
 (46) 
 
 The integral of the first of these equations may be written 
 
 R2 
 Substituting the value of '^- from the first of (23), we have 
 
 hence, 
 
 E=U^V^. (47) 
 
 From the second of equations (46) we get 
 
 This, combined with (23), gives 
 
 we have then 
 
 G = UW^. (48) 
 
 The linear element of the surface may now be written 
 
14 Certain Partial Differential Equations 
 
 or 
 
 ds' = X ( Udu' + Vdv') , (49 ) 
 
 the U's being functions of u alone and the V's of v alone. Then the lines 
 of curvature form an isothermal system. 
 
 The equations (28) both have now the form 
 
 ?<P "^(^^^P^^^f — o (K0\ 
 
 which admits — and — as particular integrals. It may also be written 
 
 Since E-=.XXJ and G=.IV , we have 
 
 dudv dv du du dv ' 
 
 The equation satisfied by [x, y, z) of any point (w, v) of the surface is now 
 
 aVo , 3 Jog \// 3^0 , 3iog\/^ a^ _ Q ,g2^ 
 
 If we have a linear equation 
 
 ai*at;+"au + ^^+'^=^' 
 
 its adjoint will be of the form 
 
 d'd dd ,dO,/ da db\. ^ 
 
 In equation (51) we have 
 
 _aa_a6_ 3Mogv>i_a^iog;._Q 
 
 *^ du dv dudv dudv ' 
 
 hence, 
 
 since ^ = f7j Fj . Then it follows that (52) is the adjoint of (51). 
 
Connected with the Theoey op Surfaces. 16 
 
 Therefore, if u and v are the parameters of lines of curvature of a 
 surface and the linear element can be reduced to the isothermal form 
 
 ds^ = k{Udu' 4- Vdv-"), {k= U^V^) 
 
 then the Cartesian coordinates {x, y, z) satisfy a differential equation which 
 is the adjoint of that satisfied by the reciprocals of the principal radii of 
 curvature. 
 
 The invariants of (51) are 
 
 Therefore, 
 We can find 
 
 dudv 
 
 The invariants of this equation are 
 
 ^ _ 1 31ogA 31og;_^ 1 
 ^ 4 du dv *[ .->. 
 
 lc, = h. J (^^) 
 
 Then we have 
 
 hi = ki=zh=ik. (56) 
 
 Then all the invariants in the series will be equal. Therefore, we are 
 led to the consideration of only one equation, (-£"01). 
 
 5. Make now the equations {En) and (E12) identical. From equations (12) 
 we obtain 
 
 _ _ VG_dlogho^ 
 
 "u — g — ^ f 
 
 Ml dv 
 
 dv ° Bi R^ dv 
 
 Equating these coefficients, we have 
 
 9 io„V(^_aiog/io2 ,-„. 
 
 dv^''^~R;-~dv~~' (^^) 
 
 Making 6u = 6i2 gives us 
 
16 Certain Partial Differential Equations 
 
 We have also 
 
 •-^-l^^+S-^l+l")'-^^'- 
 
 Since 
 
 log ^ = log F', 
 
 (59) 
 
 "~aw R, dudv ^ R 
 
 ^ -_?„wv;^ . a\/G_a\/£' , v^aiog/io2 
 ^'—awai' ^ Ri "^aw r^ dv r, '^ r, bv ' 
 
 By equating these values and taking (58) into account we get 
 
 aMog;ioi _ d \^E 
 dudv dv Ri 
 But from (23) we see that 
 
 d a/e a* , ,^ 
 
 Hence, 
 
 ^log^/GK = 0. (60) 
 
 The integral of (58) may be written 
 
 Ri 
 and this combined with (23) gives 
 
 _ aiogV-g _p 
 
 dv 
 
 Therefore, by integrating this, we have 
 
 E= U^e-'"", (61) 
 
 the integration of equation (57) will give 
 
 log F+C7= log A02; 
 consequently 
 
 hoi=iU'V'. (62) 
 
 We may write 
 
 A., = ^=y'^. (63) 
 
Connected with the Theory of Surfaces. 17 
 
 Equation (60) can now be put in the form 
 
 ^^\og^G-\-^log V + ^ log V^— ^ logi2i = 0. 
 dudv duov * dudv oudv 
 
 But from (58) we see that the second and third terms of this equation vanish. It 
 then becomes 
 
 log^=:0. 
 
 It is evident then from (33) that 
 
 "'M — r) T> y 
 and consequently 
 
 /ifli — "'01 • 
 
 Combining this equation with (63), we get 
 
 h^ = V'^=U'V'; (64) 
 
 therefore, 
 
 V^ TV 
 
 From (23) we see then that 
 
 OU 
 The integration of this equation will give 
 
 G= 7x6-2^'. (65) 
 
 Since hot = ^oi = U'V; 
 
 we obtain by differention 
 
 d" log hoi__Q 
 dudv 
 But it has been found that 
 
 J^ log ^^ = ; 
 and by substituting this in the preceding equation we find 
 
18 Certain Partial Differential Equations 
 
 This value substituted in tiie second of equations (32) will reduce it to the form 
 
 "1 — "7? D • 
 
 Clearly we have then 
 
 /Coi — /Cos — "01 —— ""OJ ) \^"/ 
 
 that is, all the invariants in the two leading equations are now equal. 
 The linear element in this case will be expressed by the equation 
 
 ^2_g-2(£r+r)|-f/-^g.£r^^2^ Fxe'^du"]. (67) 
 
 If then M and v are the parameters of the lines of curvature of a surface and 
 if the two first derived equations in the two series of Laplace (31) are identical, 
 that is, {En)=^(Ei2)', all the invariants in the two leading equations become equal 
 to each other and the lines of curvature form an isothermal system. 
 
 6. We shall now consider the general case and make (En) = (Efy). To do 
 this it is evident that we must have the coefficients respectively identical in the 
 two equations. 
 
 We have as in (41) 
 
 Gil ==. an 5— log hoihn . . . . A<_-. i , 
 
 Qi, = 003 ^ log htJhi . . . . A<_i, 2 , 
 
 = aoj — -j^ log hiJiii . . . . hfi. 
 
 Equating these values of the coefficients, we get the equation 
 
 d log hn _ ^ d log hf. 
 
 But „ __ ^/G 
 
 " — ST ' 
 
 and ^ _ /fa ^^„VE.VG\ 
 
 These values substituted in (68) reduce it to the form 
 
 ^IogA..= J,log^+|^Iog*„; 
 
 (68) 
 
Connected with the Theoey of Surfaces. 19 
 
 therefore, since 
 
 we obtain the equation 
 
 This is the first equation of condition for the identity proposed. 
 We know that 
 
 6*1 = ^00 
 0<2 =^ ^02 ) 
 
 then making 6<i = 6,2> 
 
 gives us simply 6oi = ^02 • 
 
 Substituting the values of these coefficients, the equation becomes 
 
 which is our second equation of condition. 
 
 By linear combinations of the last of equations (12) we may obtain the 
 general coefficient 
 
 „ ^ 8«oi 8«<-i,i I 3^01 
 
 c„_Coi— ^-.... -^^yT'^ dv 
 
 + ••••+ ~d^ ^'' -Wv ^ ^ 
 
 Combinations of the first of equations (41) will give 
 
 3«oi don dai-\ _ __ d^ jQg ^\ 
 
 du du ' ' ' ' du dudv 
 
 H-3^1ogASr^Air^...VM, (72) 
 
 since, from (23), we have 
 
 ^ = --1- logV^. 
 
 It may readily be seen also that 
 dv ^ dv ^ ^ dv ~ dudv ^ B\ 
 
 + al^^«g^^ = -a4^^4- (^3) 
 
20 Certain Partial Differential Equations 
 
 by remembering that 
 
 If these values in (72) and (73) be substituted now in equation (71), it may be 
 put in the form 
 
 <^ 
 
 ^ log S + 5^l0g^-'^n^ • • • • hi-^.^ 
 
 
 ~^»^ d^ 
 
 = Cn 
 
 dudv 
 
 <— 1 
 6'' 
 
 '<'g^ + 3S5s"'g5p*"---'^' 
 
 , d log hoJiii .... ^<_i, 1 
 
 ^'' di •. 
 
 (74) 
 
 But it may be seen from the first of equations (34) that proper linear com- 
 binations will give 
 
 f— 3 
 
 (75) 
 
 If we substitute this value in equation (74), we have finally 
 
 „ « IZ. 7, h 9 log ^01^11 • ■ ■ • ^^ir-\, X 
 
 (76) 
 
 In the same manner it may be shown that in the second series of equations (31) 
 the general coefficient is 
 
 Cc 
 
 ^ I ^ h h ^ lo gApgAia. . . . ^i-1,2 
 
 Co2 -|- «o2 "■<-l,2 O02 ^ 
 
 — r -u/i h —h 3 log ^11^ ^ii 
 
 Co2 -|- /loi n,i Oo2 ^- f 
 
 (77) 
 
 by remembering the relations which exist between the invariants of the two series. 
 
 Since 
 
 if we place 
 
 Ooi — 6o2 , 
 Coi=:C(a = 0, 
 
 Cft = Cc f 
 
Connected with the Theory of Surfaces. 21 
 
 we will get after obvious reductions the following equation 
 
 h —h —b ?J2i^ — A —h —h ^i^sAe nsi\ 
 
 '''01 "i-l, 1 Oqi ^-- /loi "il Ooi pC . \lO) 
 
 But we know 
 
 z. ^ _ a' , V-E^ 
 
 "'01 "01 ^5 — ^5~ log —fY- } 
 
 and from (34) 
 
 ^<l ^i-l, 1 = g^ log ^- Aii/l2i .... ^<_i, 1 . 
 
 Putting these values in equation (78), it may now be written 
 
 _. siogAoi , , a log /til _^ ,70. 
 
 On account of the second equation of condition (70) we have 
 
 and from the first condition for the proposed identity we found 
 
 a,log/^a = |,log^. 
 
 Introducing these values in equation (79) above and making slight reductions, 
 it becomes 
 
 Since 
 
 the last equation may now be written in the form 
 
 ^log VG^Wn A<-i.i, (80) 
 
 which results from making the coefficients Cn and c^g identical. 
 
 The integration of the second equation of condition (70) will give 
 
 log^ = log7', 
 
22 Certain Partial Differential Equations 
 
 or 
 
 = v. 
 
 B2 
 
 But from (23) we have 
 
 hence 
 
 By integrating the last equation we get 
 
 E=U,,-'\ (81) 
 
 The equation 
 
 3^1ogA„ = ^log-^ 
 may be integrated and placed in the form 
 
 log K, == log "^ + log W = log V + log W , 
 
 therefore 
 
 h,,= V'V'. (82) 
 
 Equation (80) may be written as follows : 
 5^ log V<? + A- log V J5^- 5^ log i2, 
 
 Bat it has previously been shown that the second and fourth terms of this 
 equation are equal to zero. It then becomes 
 
 From (34) we know that 
 
 ^ii = Vu — a^log-^— 3^1og/tiAi....Ai-i.i. 
 Hence, by combining this with the equation above, we get 
 
Connected with the Theory of Surfaces. 23 
 
 But we have found 
 
 consequently 
 
 h,^,,,= U'V'. (84) 
 
 The differentiation of the last equation will give 
 
 gf3,,l«gA-M=0. (85) 
 
 and, if this be substituted in (83), we have 
 
 Again 
 
 Ai-M='^*-2,l-a^-^l0g-^-g^^l0gM21. . . .^i-2.i; 
 
 and, consequently 
 
 Comparing this with equation (85), we have 
 
 5^1ogAi_2,i = 0. 
 
 (juov 
 
 Substituting, as before, (86) now beco nes 
 
 
 (87) 
 
 Thus, it may easily be seen that by continuing this process, annuling one 
 term each time, we are left finally with the equation 
 
 ^^ log ^c^r- = . 
 dudv -til 
 
 Then we have 
 
 "-11 ^ "-01 ^^^ "'02 • 
 
 If we should make the equations (ii^i) and {Eyi) identical, it would give 
 
 "01 =^ "'11 f 
 
 Kqi = kii . 
 
24 Certain Partial Differential Equations 
 
 But from the relations of the Id variants in the two series we have 
 
 kii = kfjfi f 
 hence A02 = ^01 = ^n > 
 
 »02 = nJoi ^^^ "-11 • 
 
 Then, by making the equations (^„) and [Ea) identical, we find that all the 
 invariants in the two series are equal and we are led only to the consideration of 
 
 Since 
 we have 
 
 
 ^log'^=0. 
 The integrals of these two equations may be written in the form 
 
 V^ —Tjv 
 
 Substituting the values of ^^— and ^— from equations (23), we get 
 
 ■til -"2 
 
 — ^ ^^ — TTV 
 
 1 ^^=Cr,F2. 
 
 The elimination of — ^=^ from these equations will give 
 
 2v^6r 
 
 or V-, dE_V^ dG 
 
 Ui dv'~ Vi du' 
 
Connected with the Theory op Surfaces. 25 
 
 where the U'S are functions of w alone and the VS ofv alone. 
 Write now 
 
 then the equation above becomes 
 
 |(ra) = |(FG). (89) 
 
 But this is a necessary condition for the existence of an exact differential equation 
 
 {UE)du-{-{VG}dv = d(py (90) 
 
 where 
 
 r — 1 ^ 
 
 Take u and v arbitrary and ^ will have to determined. 
 From (23) we have 
 
 and, by introducing the value of E from the first of the two equations above, it 
 becomes 
 
 s/G a / 1, ^, . 1. d<p\ 
 
 In like manner we can find 
 
 3m 
 
 A^E 1 dudv . 
 
 do 
 The products of these two equations will give 
 
 
 ^=iW = ^^'^>^-- m 
 
26 Certain Partial Differential Equations 
 
 If we place 
 
 we have then 
 
 W 
 
 2 fuUi-'elu+ 'ifvVi^dv 
 
 
 Hence, equation (91) may now be placed in the form 
 
 \dudvj 8 log IF 3 log W 
 
 d<pd<p 
 
 du 
 
 or 
 
 "^ ^ du' du ^ dv du ' dv 
 
 7. "We propose now to consider the result of making 6oi = «08 = 0, 
 that is 
 
 du ^ B, ^ B, ""^ 
 
 d 1 \^jE f^G - 
 
 Wv'^'^'b; -^ -B 
 
 (92) 
 
 (93) 
 
 By substituting the values found in (23) the two equations above easily 
 reduce to 
 
 |logiJ,=0, 
 
 |logiJ.=,0. 
 
 The integrals of these equations are 
 
 We may, therefore, write 
 
 B, = f{v),\ 
 B, = W{u).i 
 
 1 _ 1 dVG_.j 
 B~A^EG du ~ ' 
 
 1— l_ d^/E_.r 
 
 (94) 
 
Connected with the Theory of Surfaces. 27 
 
 where ?7 is a function only of u and V a function of v only. By eliminating 
 we obtain the equation 
 
 s/£:g 
 
 y d s/G jjd f^E 
 
 du dv ' 
 
 ^{V^G) = -^{U^E). (95) 
 
 We see that this is a condition for the exact differential equation 
 
 (96) 
 where we have 
 
 We find as before 
 
 U^E) du -\-{V^G)dv = d(p, 
 
 a/E=z 
 
 1 
 
 U 
 
 d<p 
 du' 
 
 VG = 
 
 1 
 
 r 
 
 d(p 
 
 -"2 
 
 — 
 
 dudv 
 df ' 
 du 
 
 ^E 
 B, - 
 
 — 
 
 dudv 
 df ' 
 
 ev 
 
 itial equation 
 
 
 
 dudv 
 
 d<p 
 du 
 
 d<p 
 dv 
 
 (97) 
 
 The integral of this may be written 
 
 ^ = -log(C^,+ FO. (98) 
 
 It may then easily be seen that 
 
 E= f^' 
 
 2) 
 
 The linear element may now be expressed by the equation 
 
 ^^= iU,+ v,Y [^ ^^' + -W ^"'] ^^^^ 
 
28 
 
 or 
 
 Certain Partial Differential Equations 
 
 ds'z=:^{Udw'-\-Vdi^). 
 
 A 
 
 This is a form which characterizes isothermal systems. 
 
 Then, if u and v are the parameters of lines of curvature of a surface, and if 
 the radius of geodesic curvature of the line w = const, is a function of w alone and 
 that of v=z const, of v alone, the lines of curvature of the surface will form an 
 isothermal system. 
 
 The leading equations of the two series now become 
 
 while that satisfied hy (x, y, z) remains 
 
 Since ^X=Ui-j-Vi, 
 
 we obtain a^G d log^/E F{ 3 log a,JX 
 
 A/E__d\o gVG _ U[ _aiogVA 
 
 jSx ~ au ~ u^+v~ du ' 
 
 By substituting these in the equations above we get 
 
 / r \ _ ^<foi d log A^X 3y7oi _ ~ 
 
 (j^)^ ^9 I a log v-^ a^i aiog/y/i^ ?^=zo. 
 
 ^ dudv dv du du dv 
 
 The invariants of the two proposed equations are now 
 
 »0l ^ " > 
 
 Ao2 = 0> 
 
 
 log>v/>^ 
 
 
 (100) 
 
 (101) 
 
 (102) 
 (103) 
 
 (104) 
 (105) 
 
Connected with the Theory of Surfaces. 29 
 
 That is, they have their invariants equal but taken in opposite order. 
 
 8. Next we shall consider the result of making the coefficients of the equa- 
 tions (£"01) and (£"02) equal but taken in opposite order, that is 
 
 Ooi ^^^ "02 > 
 (l02 — - ^01 • 
 
 By substituting their values we have 
 
 -^ ^""^ R^^ B^— dv ^ R, ^ R, J 
 
 (106) 
 
 (107) 
 
 By combining these we get 
 
 du ^""^ R, - dv '""^ B, ' 
 
 or _^_9 VG^ — ^ a VE 
 
 sfQ du R., ~A/Edv R^ ' 
 
 Taking into account the first of equations (106), we have 
 
 8 a/G_ d ^E 
 du R2 dv Ri 
 
 These values placed in equation (107) will give 
 
 >5^ log A^E=l -^ log VC? . 
 
 If we integrate this equation, we shall get 
 
 E= G .U.V, (108) 
 
 where ?7is a function of w only and F of v only. The linear element can now be 
 put in the form 
 
 ds'=G{UVdu'' + dv'), 
 
 or ds'=GV{Udu^-\-V,dv''). (109) 
 
30 Certain Partial Differential Equations 
 
 Then the lines of curvature form an isothermal system. 
 
 9. Let us now consider the forms to which the equations (£"01) and (£'02) reduce 
 for surfaces parallel to the original one. 
 
 Since the surface is referred to its lines of curvature we have the following 
 values for parallel surfaces: 
 
 pt = p2 — a,( 
 
 (110) 
 
 (111) 
 
 where a is a constant which changes as we pass from one surface to the next one. 
 We have now 
 
 a ^ =f p' \'d 1 
 
 du Pi \pl-{-aJ du p\ ' 
 do pi \pi-{-aJ dv p\ 
 
 (112) 
 
 By differentiating the first of these equations for v and the second for u we get 
 
 ^ 1 _r pi V a' 
 
 1^3 
 
 \p\ + a) 
 
 _a_i 1 
 
 L^( pi\ 
 
 dudv pi \pi-\-aJ dudv p^i ~* du \pt-^aj dv />?. 
 
 (113) 
 
 By substituting these values from (112) and (113) the equation 
 
 A_I__^Ai rAiog^4-VAAl. = o 
 
 dudv pi E2 du pi \du ^ B2 Ml J dv pi ' 
 can be put in either of the two following forms : 
 
 dwd» p\ Life dv ^\t>': + a)jdu f,', 
 
 \du ^ Ei ^ Ri J dv p\ 
 
 ^ 
 
 '/>? Ri du />? Idu ^~B2 ^ Bi 
 
 _-#-log 
 
 du "= 
 
 \pl + aj] 
 
 dv pl~ 
 
 (114) 
 
Connected with the Theory of Surfaces. 
 
 31 
 
 By examining these we see that in order that these equations reduce to the 
 same form as the original equation, we must have 
 
 The integration of these will give 
 
 (115) 
 
 (116) 
 
 In the same manner as above the equation (£"02) may be put in either of the forms 
 
 d^v f^ L dv '""^ R; ^ R, dv '""^ \{,l + a) J du pi 
 
 J 
 
 dud 
 
 ^v pi V dv ^ R, ^ R, J du pi 
 
 VE d_ JL 
 R^ dv pi 
 
 = 0, 
 
 iR, du'^'^Kpl + aJ] dv pi 
 The condition that these reduce to the same form as (i/oa) is, as before. 
 
 (117) 
 
 pl = (p{u), 
 
 = (p{u), I 
 = (pi{v).S 
 
 (118) 
 
 Then if each of the principal radii of curvature of the parallel surface is a func- 
 tion of only one of the parameters u and v, the reciprocals of those radii will 
 satisfy the same partial differential equations, (£"01) and (£"02) , as the reciprocals of 
 the principal radii of curvature of the original surface. 
 From (110) we have 
 
 T' ^ 
 
 Then, the equations 
 
 Pi P 
 
 \/Gr \/(to 
 
 pi 
 
 d 1 yjg a VE s/EG ^E^ Q ^ 
 
 dudv pi du Rz dv pi R1R2 ' Pi 
 
 _?_ V<^ d^ s/GdVG \^EG yG_^ 
 
 dudv Pi dv Ri du p2 RiRi ' p2 ' 
 
32 Certain Partial Differential Equations 
 
 do not change form when we pass to a parallel surface. 
 
 10. (a) For the ellipsoid we have ^='i£-^;::;^, ^— ^(^— ^)^ Then 
 equations (^oi) and {E(^) become 
 
 ^udv 2{u — v) du 2{u — v) dv ' 
 
 ^02_ 3 9f02 I 1 9f02__Q . 
 
 dudv 2 (w — v) 3w 2 (w — v) dv ' 
 
 (119) 
 
 and the equation satisfied by {x, y,z)i8 
 
 a^^o , 1 d(po_ 1 d<po--Q^ Q20) 
 
 dudv '2{u — v) du 2{u — v) dv ' ^ ^ 
 
 We can obtain a great number of particular integrals of these equations. 
 For example, we can find solutions which are the product of a function of w by a 
 function of v. 
 
 If a denotes a constant, we write 
 
 ^01 = (w — a)-^ {v — a)-i 
 
 1 
 
 \/{u — af {v — a) ' 
 <p^=:{u — a)-^{v — a)-i 
 1 
 
 Ai/{u — a){v — a)'^ 
 By using a known property of these equations we have also 
 
 ^01 = (^ — u)~^(u — a)~^{v — a)* 
 
 V — w T u — a 
 
 ^02 = (v — u)-^ {u — a)i {v — a)-* 
 1 l u — a 
 
 V — u y V — a* 
 
 "We can find new solutions by operating on those already found with the symbols 
 
 -'i + '^li + i^ + i"- 
 
Connected with the Theoby of Surfaces. 33 
 
 We have also the general formulae 
 
 ' f{x){x — u)-^ {v — x)~ '^ dx-\-{v — u)~ 1 / /i {x){x — .w)~ ' (v — «)* dx , 
 
 u vu 
 
 f{x)[x — w)~* {v — x)~ i dx -\-{v — u)~ 1 / /i {x){x — w)* [v — x)~ ^ dx , 
 
 which contain two distinct arbitrary functions. In the same way we can obtain 
 integrals of equation (120). The integration of this equation will determine the 
 surfaces which admit for spherical representation of their lines of curvature a sys- 
 tem of spherical homofocal ellipses. If u and v denote the parameters of these 
 ellipses and X, [x, v the coordinates of a point of the sphere, we have 
 
 ,2_(a--w)(a--v) 
 '^ — (a — 6)(a — c) ' 
 
 (6__w)(6--j^) 
 ^ — (6 — a)(6 — c)' 
 2 _ (c--w)(c— j;) 
 — (c — a){G — h) ' 
 
 and the values of ^, fx, v will satisfy equation (120). The solution 
 
 ^0 = C-J-C'(M + V), 
 
 will give the surface of the fourth class defined by the equation 
 
 a)? -j- b[j? -|- ei^ 
 
 P— 2 ' 
 
 where ^, //, v are the direction cosines of the normal, and the coordinates of the 
 point of contact of the tangent plane have the values 
 
 x = {p^a)X, y=:{p—h)fx, z=z{p — c)v. 
 If we take the solution 
 
 ^0 = G\/{u — a){v — a) f 
 
 we obtain the surfaces of the second degree. 
 
 Equation (120) also admits as particular solutions the five pentaspherical 
 coordinates of the cyclides defined by the formula 
 
 ^^^^ (a.-a)(«.-«)(a.-.) _ (^^ i, 2, 3, 4, 5). 
 
34 Certain Partial Differential Equations, etc. 
 
 (b) For the cyclide of Dupin the coefficients of the linear element are 
 
 G = 
 
 1 
 
 ^V{u — v)J' 
 The equations {Eqi) and (jS'02) now become 
 
 3Voi I 1 dfo} 
 dudv u — V du 
 
 CPJPo 
 
 = 0, 
 
 1 3f02__Q 
 
 dudv u — V dv 
 while that satisfied by the point coordinates is 
 
 (121) 
 
 (^0) = 
 
 3^0 _ 
 
 dudv u — V du u — V dv 
 
 = 
 
 (122) 
 
 We can obtain any number of algebraic integrals of these equations ;^r 
 example 
 
 >oi = V — a , 
 
 foi 
 
 y?02 = w — a, 
 [u — vf 
 ' (v — a)[u — af ' 
 
 {u — vf 
 
 'P^ — {u — a){v — af' 
 
 The integration of (J^o) gives 
 
 ^0 
 
 U- 
 
 In fact, by making the substitution 
 
 ^0 = 
 
 M V 
 
 U V 
 
 the equation {E^ is transformed into 
 
 dudv 
 
LIFE. 
 
 I was born at Loachapoka, Ala., and received my early training at the High 
 School at that place. I was graduated from the Southern University (Ala.) with 
 the degree of Bachelor of Science in 1887. After spending one year as principal 
 of a high school, I returned to my Alma Mater, where I remained two years as 
 Instructor in Mathematics, and received the degree of A. M. in 1890. Having 
 spent another year teaching, I entered the Johns Hopkins University in 1891, 
 where I pursued graduate courses in Mathematics, Physics, and Astronomy. 
 During 1892-'94 I was Professor of Mathematics in Millsaps College (Miss.). In 
 1894 I returned to this University. I have attended the lectures of Professors 
 Craig and Franklin and Drs. Ames, Poor, Hulburt, and Chapman, to all of whom 
 I am grateful for the kindnesses that I have received from Jl 
 
 AprU, 1897. 
 

 MAY 2 1935 
 
 MAR 21 1! 
 
 
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