354 REPORTS ON THE STATE OF SCIENCE, ETC. 
The solution of (45) has been obtained by trial, and is exhibited in tig. 17. The 
procedure adopted is to trace a contour line (T=const.) on a diagram of which the 
co-ordinates are P and w?. We are then, by (43), given the value of (@+e), but can, 
with this limitation, vary the separate values : choosing any pair of values, we have 
from the third of (43) a range of corresponding values of b” and d*, ‘The sign is 
immaterial, and for each pair we calculate the left- and right-hand expressions in 
(45). Proceeding in this way by trial, when the expressions are equal we know 
a, b, ¢, d, and hence B and C, from (43) 
It may be noticed that a possible solution of the criterion (45) is given by a=c=O, 
2d=nmr. This gives 
A=O, 
B = a? + Crake 
4 
eS C. 
4 
and hence 
ee + 16C a 
nx n't 
or 
Iie bs Wowlt _ 
weEl atetgEl 
which reduces to (18) when »=1. When 'T is small, therefore, we may expect a 
solution round about the value 2b=~y7, and for practical purposes 7 will be 1. 
§9. Solution for Clamped Ends.—Employing as before the substitutions of 
(41)-(43), and multiplying (40) throughout by 2 we may write this criterion in 
the form 
(Hy — Hz) (Mg — By) COS (My + My My — Hy) 
— (Hy Ms) (Hy — By) COS (oy + My — My — My) 
= (My — My) (Hy — My) COS (My + My — My — Hy’ 
a = 1 (Hy = M2) (Hy — by) — (ty — My) (Hp — Hy) | COS (My + Hg — Ba Fa) 
(Hy ~ My) (My Hy) Sin (My — My) SID (My — My) 
= (Hy — My) (My = My) SiN (ty — My) SID (My — My), : : - (46) 
which corresponds with equation (44) of the last section, Hence, if we substitute 
from (42), we obtain the relation 
4bd sin (a + b—c—d) sin (a—b—c+d) 
=(a4+b—e—d) (a—b—c+d) sin 2b sin 2d, 
or 
2bd {cos 2(b—d) —cos 2(a—c)} 
= {(a—c)?—(b—d)?} sin 20 sin 2d, 
which may also be written in the form 
2bd {cos 2b cos 2d —cos 2(a—e)} 
={(@—c)?-b?—d*}sin2dsin2d, . : : . (47) 
for convenience in dealing with imaginary values of 5 or d. 
ee (47) has been solved, like (45), by trial, and the results are exhibited 
in fig. 18. 
= —— 4 ¢ ae r 
