LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class - HIP AND VALLEY LAY-OUTS AND TABLES FOR ENGINEERS AND ARCHITECTS BY A. H. T. WILLIAMS AND W. H. CULLERS FIRST EDITION 12.00 POSTPAID COPYRIGHTED IQIO W II. CULLER Nurtftwst Steel Co. Portland, 0. OFTHE UNIVERSITY PREFACE Portland, Oregon, August 1st, 1910. In the absence of any definite information relative to hip and valley work, and knowing the great difficulty with which many experienced as well as student draftsmen attack this special line of structural work, the authors submit these plates and tables to the structural draftsmen. We have included at least ninety per cent, of the problems usually met with in this class of work and believe we have made it possible for the ordinary draftsman to easily lay out and detail the greater majority of connections that confront them in roof work. The first plate showing the method of getting the general figures necessary for detailing, while apparently simple, has often been the downfall of the beginner in hip work, and as a proper start is the most important feature of any kind of work, we believe that this drawing is very necessary. The remainder of the plates are so laid out that the draftsman in carefully following the lines of projections will have no difficulty in meeting all the conditions that arise in these complex problems. As the purlin connection, shown on Plate 2, normal to the roof line and flush on top, is most generally met with, we have taken it as standard and have tabulated all the necessary bevels for detailing this connection for a roof starting with a bevel of three inches in twelve and ending with eight and fifteen-sixteenths inches in twelve so that this class of joint may be detailed without the laborious work necessary in finding" the bevels, thereby reducing the layout work to a minimum. X 210633 to PLATEL2. CONNECTION OF- PUGL/M I 3 ^ PLATE 4. WEB AND FLAHGZ CONNECTIONS ro PLATE 5. FLANGE CONNECT/ON or TEE PU&LIN AND Htf* PL.ATEG. WEB CONNECTION or HORIZONTAL BEAM TO HIR PLATE 7 l > > Q tf 5 t rj cr FLANGE CONNECTION OF PLATE B if r* WEB CONNECT/ON OF I/VCUHED TO Mf>. PLATS 9. WEB CONNECTION OF HORIZONTAL BEAM A NO TILTED n/P 0) 3 PLATE /a CONNECTION OF TILTED BEAM AND TILTED HIP. PLATE II, KNEE B&ACE CONNECT/ON OF COL VMM PLATE/2 a Angle A b c d e f g h 3 D 14 M 2 8 10 2% 2 2M ny 8 lift 1% 2A ft 14 19 2& 2& 2H ny 8 lift 1% 2% % 14 35 48 2& 2A 3 ny 8 HM 1ft 2A A 14 52 32 2M 2^ 3^ 11% nji 1ft 2A M 15 9 15 2& 2^ 3^ lift lift 1% 2M ft 15 25 54 2& 2& 3A lift lift 1% 2ft v % 15 42 31 2% 2A 3M lift n% Itt 2ft A 15 59 5 2& 2M 3& lift n% Itt 2% 3% 16 15 37 2% 2M 3^ iiH HA 1% 2& ft 16 32 5 2% 2A 3A HH lift 1% 2ft Hi 16 48 31 2ft 2% 3K HH lift 1 2% H 17 4 54 2% 2^ 3H HH 11 1 2ft % 17 21 14 2% 2A 3& lift 11 1% 2ft u 17 37 32 2& 2A 3% lift io 1% 2% % 17 53 46 2M 23^ 3H lift iott Itt m it 18 9 57 m 2^ 3M HH 10% lit 2& 4 18 26 6 2tf 2A 3M HH 10% 2 2M A 18 42 11 2% 2^ 3^ n% ion 2 2tt H 18 58 13 2tt 2^ 3^ n% iott 2ft 2tt A 19 14 12 2 2^ 3H lift 10% 2^ 2% M 19 30 9 3 2tt 4 lift 10% 2% 2tt A 19 46 2 3& 2tt 4^ lift iott 2% m % 20 1 52 3^ 2% 4^ HM iott 2ft 3 A 20 17 30 3^ 2M 4A UH 10% 2& 3A 4% 20 33 22 3A 2H 4A HM 10% 2M 3^ A 20 4<) 2 3H m 4^ lift 10ft 2M 3% % 21 4 39 3M 2K 4A lift 10ft 2ft 3ft H 21 20 13 3^ 2% 4^ lift 10% 2ft 3ft % 21 35 43 3^ 2^ 4A lift 10^ 2% 3M tt 21 51 10 3^ 2 4A n% 10ft 2% 3^ 22 ' 6 34 3A 2tt 4^ 11% 10% 2& 3A B 22 21 ' 54 3K 3 4^ 11% 10% 2A 3% a Angle A b c d e f g h 5 D 22 M 37 s 12 3ft 3 4^ HA 10ft 2^ 3^ ft 22 52 25 3ft 3^ 4tt HA 10ft 2K 3ft l /s 23 7 35 3^ 3A 4^ HA 10% 2ft 3K ft 23 22 42 3& 3^ 4^ 11 10% 2ft 3K % 23 37 46 3& 3^ 4M 11 10ft 2^ 3ft A 23 52 46 3% 33^ 4K 11 10ft 2^ 3ft % 24 7 42 3tf 3^ 4% iott 10% 2tt 3^ A 24 22 35 3% 3& 4 ion 10% 2tt 3& 5H 24 37 25 3% 3^ 5 iott 10ft 2% 3tt A 24 52 11 3 3M 5& 10% 10ft 2% 3% % 25 6 53 4 3M 5^ 10% 10 2tf 3% 25 21 32 4 3^ 5H lOtf 9tt 2tf 3}| % 25 36 8 4ft 3& 5A io 9*t 2K 3% 25 50 40 4^ 3^ 5^ IOH 9% 2% 3% 7 /8 26 5 8 4^ 3^ 5M 10% 9% 2 3 it 26 19 33 4ft 3^ 5A 10% 9H 2M 3 6 26 33 54 4% 3^ 5% 10% 9H 3 4 A 26 48 12 4ft 3& 5 -A iott 9% 3 4ft % 27 2 26 4ft 3A 5A iott 9% 3ft 4ft ft 27 16 36 4^ 3A 5K iott 9& 3A 4^ % 27 30 43 4& 3^ 5^ 10% 9tt 3^ 4^ A 27 44 46 4^ 3^ 5 A 10% 9^ 3^ 4ft % 27 58 46 4^ 3H 5^ 10% 9^ 3ft 4% A 28 12 42 4& 3A 5H IOA 9ft 3ft 4% 6^ 28 26 35 4^ 3^ 5M 10ft 9^ 3% 4ft A 28 40 23 4^ 3A 5M IOH 9^ 3% 4ft % 28 54 9 4^ 3A 5M IOH 9ft 3ft 4% ft 29 7 50 4M 3^ 5M IOK 9ft 3ft 4^ % 29 21 28 4M 3^ 5^ IOA 9% 3ft 4ft H 29 35 2 4H 3^ 5*f IOA 9% 3^ 4^ % 29 48 33 4^ 3tt 5M l<), 7 (i 9ft 3^ 4^ H 30 2 4K 3^ 6 10% 9ft 3ft 4ft Angle A D 30 M 15 s 23 314 g ft 30 28 43 6^5 10ft 30 41 59 10ft 30 55 11 5ft. 3% 10ft 31 20 3M 31 21 26 3M 31 34 27 A 31 47 25 103% 32 19 10ft 4% A 32 13 10 4% 32 25 57 3 if 32 38 41 3% 8% 32 51 21 10ft H 33 57 10ft 3% 33 16 30 10ft tt 33 33 28 59 10 8M 3% 41 24 6% 10 33 53 46 34 5M 34 18 19 9% 5ft 34 30 31 9% 34 42 38 5% 34 54 43 35 43 5ft 35 18 40 4ft A 35 30 34 9M N 35 42 24 9M 35 54 11 9M 36 54 H 36 17 33 8ft 4ft it 36 36 29_ 40 10 -42 _8_ft 8M 4ft 4^ DROP OF HIP X ROOF BEVEL WIDTH OF FLANGE 3 3^ 3M 3% V/2 3^ 3M r/ 8 4 4K 4M 4^ VA VA 4M 4% 5 5^ VA 3 M M A ft A A A A % N % ^ % N A A A A A A M H A A A A A N y* H % N H A A A A A A 1 A M & A A A A A % % N K N A A A A A A X A M A A A A A N N % N % % A A A A A y 2 1 A M A A A A A A ^ H N H */8 A A A A A K 1 A 1 A A A 1% A A A H N N N ^ A A A A A A H 1 A y 2 X A A A A A % ^ N N H A A A A A K 1 A 1 A X A A A A A N H N N A A A A A 1 A H 1 A 1 A 1 A 3K A A A A N */8 M N H A A A A A 1 A H 1 A 1 A 1 A A A A A ^ H H H N A A A A A H 1 A X 1 A y 2 A N A A A N H H N H A A A A H H 1 A H 1 A A A ft A A 3 A N % % */8 A A A A A ^ H 1 A X y* A A M A A % N N % N A A A A H ^ H 1 A 1 A A A A ft A A X N N N A A A A A K X K 1 A A A A A K A X % N N N A A A A H H 1 A H 1 A A A A A tt A H 3 A X N A A A- A A M H 1 A H A A A A A 4 N N X N ^ A A A A H H 1 A X K A A A A N A N H Z A ** A A A A A y* H 1 A 1 A A A A A X N H N N % N A A A A H 1 A ^ H A A A A A X ^ A X ** H ** A A A A H X H X A A A A X X N M ^8 N % A A A A H 1 A 1 A K A A A A A X 5 /8 X A M H H A A A A y* H M A A A A X X X X N N % x A A A A K x 1 A H A A A A X 5 /8 X X A N % A A A A H l /2 H 1 A A A A A X X X X ft 4^ N % A A A A H H 1 A y* A A A A % X 5 /8 ft ft A N H A A A H H H 1 A A A A A 5 /8 y 8 X X ft ft N H A A A A ^ 1 A K 1 A A A A A 5 /8 X X tt ft ft ft N A A A A M 1 A K A A A A 5 /8 5 /8 X X tt ft ft M N A A A K K 1 A H A A A A 5 /8 5 /8 y* H H ft ft H A A A A H 1 A 1 A H A A A 5 /8 5 /8 X X tt ft ft ft 7 /s A A A A ^ 1 A K A A A A 5 /8 5 /8 5 /8 X tt ft ft H H A A A X H 1 A 1 A A A A A y s % X H ft H M DROP OF HIP = X ROOF BEVEL WIDTH OF FLANGE 5 5j^ 5^ 5M 5^ 6 6^ 6M V/8 V/2 6^ 6M 6K 7 7 l /8 7K 7*/ 8 7K 7^ 7M 3 A H H H H y 2 A A A A A A H H X H H N ft ft A H Hi H K K A A A A A A 5 /8 5 /8 5 /8 5 /8 5 /8 ft ft ft l /s M H K H A A A A A A N 5 /8 K 5 /8 it ft ft ft ft ft A l /2 K l /2 H A A A A A 5 /8 W X N H ft ft ft ft H ft 1 A 1 A K 1 A A A A A A H X % X H ft ft ft ft ft ft A 1 A H A A A A A 5 /8 X N 5 /8 ft H i* ft ft ft X % X 1 A A A A A A 5 /8 5 /8 X H ^ ft ft ft ft ft X % X M A A A A A A 5 /8 5 /8 5 /8 5 /8 H ft ft ft ft ft X % % X M 3^ A A A A A 5 /8 5 /8 5 /8 K H ft ft ft H x X % X % if A A A A A M 5 /8 X 5 /8 5 /8 ft ft ft ft M X X X X if if X A A A H % % 5 /8 5 /8 ft ft ft ft ft X X X X if if if ft A A 5 /8 N N 5 /8 5 /8 ft ft ft ft ft M X X % if if if if M A A 5 /8 N N 5 /8 ft ft ft H ft M M X X if if if if H if A ^ 5 /8 W M ft ft ft ^ ft M H X X X if if if if K 8 % ^ 5 /8 N N ft ft ft ft M X M % X if if if if % K if % ^ 5 /8 N ft ft ft ft X M X M X if if if if H 7 /8 K 4 N 5 /8 5 /8 ft ft ft ft ft X M % X if if if if 7 /8 K 7 /8 K A H 5 /8 5 /8 ft ft .ft ft M X X X if if if if H 7 /8 % 7 /8 K H ^ 5 /8 H ft ft ft M X X X if if if if if 7 /8 7 /8 7 /8 7 /8 if A N tt H ft ft ft M X % X if if if if H 7 /8 7 /8 7 /8 if if 1 A H tt H ft ft M X X if if if if 7 /8 7 /8 7 /8 7 /8 if if if A ft ft ft ft M M X X if if if if 7 /8 H 7 /8 7 /8 if if if M % ft H ft ft X M X X if ft if if 7 /8 7 /8 7 /8 7 /8 if if if if A ft tt ft M % M X if if if if % 7 /8 7 /8 7 /8 if if if if i 4^ tt H M X 3 4 if if if if 7 /8 X H 7 /8 if if if if 1 i A ft tt fc X X M if if if if 7 /8 7 /8 7 /8 if if if if 1 1 i H ft H M % % if if if if % 7 /8 H 7 /8 if if if 1 1 1 i ft ft % M X if if if if 7 /8 7 /8 7 /8 7 /8 if if if if 1 1 1 i& M M % X % if if if if % 7 /8 7 /8 if if if if 1 1 1 1 1A H M Ye X if if if if H % 7 /8 7 /8 if if if 1 1 1 1 1A 1* H M H M' if if if J4 7 /8 % 7 /8 if if if if 1 1 1 1& i& 1A it M M it if if if 7 /8 H % if if if if 1 1 1 1 1A 1* 1A DROP OF HIP = X ROOF BEVEL WIDTH OF FLANGE 3 3H 3M &A 3K 3^ 3M 3K 4 4^ 4M 4% 4^ VA 4M VA 5 5 1 A 5^ 5 & A A y 2 H y 2 X A A A X 5 /8 X X ft ft ft M X A A A A y 2 y 2 y 2 A A A A X X X ft ft ft ft k % l /8 A A y 2 y 2 y 2 y 2 A A A X X X X ft ft ft H H H A A A y 2 y 2 y 2 y 2 A A A 5 /8 X X ft ft ft ft M M M K A A y 2 y 2 y 2 A A A A X X X ft ft ft X K K M ft A A y 2 y 2 1 A A A A X X 5 /8 X ft ft ft M H % it X A y 2 y 2 y 2 y 2 A A A X X X ft ft ft M M M it A A y 2 y 2 y 2 A A A A X X X ft ft ft X H X it it 5K A y 2 y 2 y 2 A A A X X X X ft ft ft X X M it it A A y 2 y 2 y 2 A A A X X X ft ft H M % M M it it 5 A y 2 y 2 y 2 y 2 A A A X X X ft H ft H H M it it it ft y 2 y 2 y 2 A A A X X X X ft ft ft M H X it it it M y 2 y 2 y 2 A A A X X X ft ft ft M X H it it it it H 1 A H 1 A A A A X X X ft ft ft X H X it it it 7 A K y 2 y 2 A A A A H X X ft ft ft % H H it it it 7 A it y 2 y 2 A A A % ^ X tt ft ft M H % it it it 7 A 7 A 6 y 2 y 2 A A A X % X H ft ft H X H it it it 7 A 7 A A y 2 H A A A X K X ft H ft M % H it it it H 7 A H y 2 y 2 A A X 5 /8 K tt ft ft M M H it it it 7 A 7 A 7 A A y 2 A A A X X H tt ft ft X M % it it it 7 A 7 A 7 A X y 2 A A A 5 /8 X K tt H ft X M % it it it 7 A 7 A it A y 2 A A A 5 /8 X X tt ft H H X it it it H 7 A 7 A it H y 2 A A A H 5 /8 tt tt ft X X H it it it 7 A 7 A 7 A it A A A A ** 5 A 5 /8 ft tt ft % % % it it it 7 A H it it 6H A A A 5 /8 H X ft tt ft H % it it it % 7 A 7 A if it A A A A 5 A 5 A 5 /8 tt H H H % it it it J4 7 A 7 A it it N A A A 5 A 5 /8 tt tt tt H H H it it it 7 A 7 A if it it H A A 5 /8 5 A 5 A ft H tt H X M it ft H 7 A 7 A it it it M A A 5 /8 5 /8 5 /8 tt tt tt X M H it it % 7 A 7 A it it i H A A % 5 A 5 A tt tt M H M if it it K 7 A it it ft i 7 /8 A A 5 A % H to tt M H X it it X K 7 A it it it i H A A 5 /8 X tt ft H M M H ft it H % 7 A it it i i DROP OF HIP = X ROOF BEVEL WIDTH OF FLANGE Sfcj 5K 5K 5M 5K 6 6K 6M ^^ 6K 6K 6M 6K 7 7K 7 1 A 7M 7M 7K 7M 5 K H if if if % 7 A K % it it it 1 1 l 1 1ft ift 1A IK & M if if if if % 7 A K it it it it l 1 l 1A 1A 1A 1A IK K M if if if 7 A K 74 7 A it it it 1 l 1 l 1A 1A 1A IK IK ft if if if if 7 A K 7 A it }^ it it 1 l 1 1ft 1A 1A IK IK IK X H if if M 7 A K 7 A it it it l 1 l 1A 1A 1A 1A IK IK IK A if if if 7 A 7 A K it it it 1 l l l 1A 1A 1A IK IK IK ift H if if H 7 A 7 A % it it it 1 1 l ift 1A 1A IK IK IK IK ift A if if 7 A H 7 A it it it l 1 l 1 1A 1A 1A IK IK IK ift ift 5K H H H 7 A 7 A it it it l 1 1 ift 1A 1A IK IK IK IK ift ift A if 7 A % 7 A it it it 1 i 1 ift 1A 1A 1A IK IK IK ift ift ift *A % % % 7 A it it it 1 i 1 1A 1A 1A IK IK IK ift ift ift IK tt K H % it it it l l l 1ft 1A 1A 1A IK IK IK ift ift ift IK M K 7 /s % it it it i 1 i 1A 1A 1A IK IK IK ift ift ift IK IK if % 7 /8 it it 15 16 1 i 1 1A 1A 1A 1A IK IK IK ift ift ift IK IK K % 7 /S it it it l i 1 1A 1A 1A IK IK IK ift ift ift IK IK IK H 7 /s it it it 1 l l 1A 1A 1A IK IK IK IK ift ift ift IK IK IK 6 7 A it it it 1 1 i 1A 1A 1A IK IK IK ift ift ift 1M IK IK ift A 7 A it it it 1 1 l 1A 1A 1A IK IK ift ift ift 1M 1M IK ift ift K it it it i 1 l 1A 1A 1A IK IK IK ift ift ift 1M IK IK ift ift A H it it i 1 1 1A 1A 1A IK IK IK ift ift 1M 1M IK ift ift ift K it it i i 1 1A 1A 1A 1H IK IK ift ift ift 1M 1M IK ift ift ift A it it i i 1 1A 1A 1A 1H IK IK ift ift 1M 1J* iJi ift ift ift IK H it it i i 1A 1A 1A 1H 1H IK ift ift ift 1M 1M 1M ift ift ift IK A it 1 i i 1A 1A 1A 1H IK IK ift ift 1M 1M 1M ift ift ift IK IK 6K it 1 i i 1A 1A IK 1H 1 1 A ift ift ift 1M 1M 1M ift ift ift IK 1H ft i 1 i 1A 1A 1A IK 1H ^A ift ift ift 1M 1M ift ift ift 1% IK IK K i 1 i 1A 1A 1A IK 1H ift ift ift 1M 1M 1M ift ift ift IK IK ift H i 1 i 1A 1A IK IK 1H ift ift ift 1M 1M ift ift ift IK 1% 1% ift M i 1 ift 1A 1A IK IK 1H ift ift 1M 1M 1M ift ift ift IK 1% ift ift i 1 VA 1A IK IK IK 1A ift ift 1M 1M 1M ift ift IK IK IK ift 1A ^ i 1* iA i-A IK IK IK ift ift 1M 1M i& ift ift ift IK IK ift ift ift *i i 1* 1A 1A IK IK ift ift ift 1M iJi 1M ift ift i*A 1H IK ift ift ift DROP OF HIP = X ROOF BEVEL WIDTH OF FLANGE 3 3K VA V/8 3K 3^ 3M y/8 4 4^ 4M 4^ 4K 4^ 4M 4% 5 5K 5M 7 * % 5 /8 5 /8 tt tt tt *A X it it it 7 /8 7 /8 X it it 1 1 A A 5 /8 5 /8 5 /8 tt tt X M 3 A it it it X % it it it 1 1 K ft 5 /8 5 /8 5 /8 tt tt H M y it it 7 /8 H X it it it l 1 A A 5 /8 H tt tt tt M M y it it X 7 /8 7 /8 it it 1 l 1 X A 5 /8 H tt tt tt X M it it it X X it it it 1 1 ift A A 5 A H tt tt tt M *A it it it % 7 /8 it it it 1 l ift X N 5 /8 N tt tt H M X it it % H 7 /8 it it 1 1 l 1ft A H 5 /8 H tt tt M M X it it X X 7 /8 it it 1 1 l 1ft 7K H 5 /8 tt tt tt H ^ it it it H % it it it 1 1 1ft 1ft A y* X tt tt tt H M it it it 7 /8 % it it it 1 1 ift 1ft H 5 /8 5 K tt tt tt H M it it X X Y* it it i 1 1 ift 1ft H 5 /8 H tt tt H M M it it X X % it it i 1 1ft 1ft 1ft M X_ 5 /8 tt tt H M M it it X X it it it i 1 1ft ift 1ft it 5 /8 5 /8 tt tt X M it it it X X it it it i 1 1ft ift IK K 5 /8 tt tt tt M H it it it 7 /8 X it it 1 i 1 1ft ift IK it 5 /8 tt tt tt H M it tt X X X it it 1 i 1ft 1ft ift IK 8 5 /8 tt tt H X M it it X 7 /8 7 /8 it it 1 i 1ft 1ft IK IK & 5 /8 tt tt *A H M it it X X it it it 1 i 1ft 1ft IK IK H 5 /8 tt tt X % it it it X 7 /8 it it 1 1 i 1ft 1ft IK IK A H tt tt X M it it it X X it: lit 1 1 ift 1ft 1ft IK IK 1 A 5 /8 H tt H H it it X X 7 /8 it 1 1 ift 1ft 1ft IK IK & tt tt tt H % it it 7 /8 7 /8 it it it 1 1 ift 1ft 1H IK IK H H tt H % M it it X X it it it 1 1 ift 1ft 1M IK ift A tt tt X X M it it X X it it 1 1 1 ift 1ft 1H IK ift 8K tt tt X X tt it it X X it it 1 1 1ft ift 1ft 1H IK ift & tt tt X % if it H X 7 /8 it it 1 1 1ft ift 1H IK IK ift H H tt % *A it % % H it it 1 1 1ft ift 1^ IK ift ift tt H tt X M it it X X it it it 1 1 1ft ift 1H IK ift ift 9* H tt H it it 7 /8 X it it 1 1 1 1ft ift 1H IK ift ift H tt X % M it it X % it it 1 1 1* 1ft IX 1H IK ift ift % H X H tt it it % X it it 1 1 1ft 1ft V/8 1M ift ift ift it H H % it it H % X it it 1 1 1ft 1ft V/8 1>'8 ift ift IX DROP OF HIP ROOF BEVE WIDTH OF FLANGE IA tt H it A A H H A IK iK IK 1A 1A 1A 1A 1A iA 1A IK IX 1A 1A I _1M ^K JK IK IK IK IK iA 1A 1A iA 1A 1A IA JA IK IK IK IK IK IK IK Jjfe ^K ^K ^K ^K ^K JK IK IK 1A 1A 1A iA A A A IK IK IK IK IK IK IK |K _1K |K K ^K ^K ^K 1A 1A ^A ^A 1A IA IK IK IK IK IK IK J^ IK ^K ^K ^K 1A A i* 1A 1A 1A IK IK IK IK IK IK IK IK 1A J^ ^K A iA ^A i& i& A ^K iK ^ iK iK IK iK IK 1A 1A 6K TS ^A 1* 1* A 1A 1M ^K IK IK IK iK IK 1A 1A 6K TS 1A ^K J34 ^K ^K K IK IK 1A 1M 1A 1A _1A ^K Jr ^K ^K ^K ^K IK 1A 1A 1A 1A 1A 1A ^K ^K ^K _1K IK 1A 1A 1A 1A IK ^ IK 1A 1A iA _1A 1A IK 1A 1A 1A 1A 7K iA IA ^A 1A ^A 1A 1A 1A iA 1A 1A 1A 1A 1A ^A ^A 1A ift itt !$_ ^tt itt J_tt U* itt ^A 1* ^A IA 1A ^tt Ui JJ* jit ^tt 11* itt /.- 4%, %-' UNIVERSITY OF CALIFOENIA LIBRARY BERKELEY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW expiration o'f loan period. Id MAY 3 U.C.BERKELEY 50m-7,'16