372 Prof. E. Wilson. [May 28, 



the form a or ft being taken according as ma is less or greater than 



Hence the rate of dissipation of energy in watts per cubic centimetre 

 is A + B, where 



a*/ (I— m 2 ) ma 



A 



16tt/ 2 B 2 f 



mcv 



d f W-a*)-ma + z)W 

 3 /°10 16 J ft J '( </(a 2 - x 2 ) -ma + 2z) az ' 



o "0 

 a V(« 2 -* 2 ) 



, _ 1 6tt PW f , f (M-v/(a 2 -« 2 ) + y) 2 v 2 , 



and = ^j fl ,;! m2) j f^tfw* • 



When m is greater than unity the integral B alone is taken, the 

 limits of x being and a. 



To integrate A with respect to z, let 



J (a 2 - x 2 ) -ma = 2D and z + D = £. 



Then 



ma Kia+D ma+D 



[ W(a?-x*)-rm+sYz* , _ f (g 2 -I>M _ if? 4 . op , # w/1 

 J (y(a>-x*)-ma + 2zr-\ 2? 2 \J £ + ° S *J' 



DO 



or, restoring the values of D and £ 



A ma ( v/(a 2 - z 2 ) ) (ima v'(a 2 - x 2 ) - ra% 2 - (a 2 - x 2 ) ) (C) 



+ ~(^/(a 2 - x 2 ) - may log ^f'^ +WIA (D). 



A similar substitution in B yields as the result of integration, with 

 respect to z, 



\ ma J (a 2 - x 2 ) (4ma J {a 2 - x 2 ) - m 2 a 2 ~(a 2 -x 2 )) (C) 



+ i (» - ^ 2 - * 2 ) ) 4 ^g (i>')- 



2 5 v v \mft - v/(^ 2 - a; 2 )/ 7 



Since (C) and (C) are identical in form, we can combine them, and 

 integrate with respect to x from x = to x = a. 



Let £ = a sin 6, and denoting cos 6 by c and sin 6 by 5, we obtain 



2 4 \3 16 



o " o 



a ir/2 



^(C + C')dx = c 2 (4mc-c 2 -m 2 ) 



(E). 



By the same substitution D and D' assume the forms 



cos ~ 1 ra 



j> :;: i .(,• " ./.- (F), 



