Doctrine of Extraneous Force. 127 



Now we have, by (2), log (a/3y)=0. Hence, taking the 

 variation of this as far as terms of the second order, 



h + M + h -i^ + ^ + ^1=0 (49) • 



a + + 7 2 U 2 + /3 2 + 7 2 , J ° ' ^' 



which reduces (48) to 



/ s* g/3 s 7 s* 2 w s y 2 \ , 



SE = 2 H~^-F~7 T+ ^ + F + 7 3 > ) • (50) " 

 Remembering that cubes and higher powers are to be neg- 

 lected, we see that (50) is equivalent to 



SE 



=*< + H) (51) - 



Hence if we take k constant, we have 



B -**(W + ?-*) • • • • (52) 



and it is clear that k must be stationary (that is to say, Sk=0) 

 for any particular values of a, /3, 7 for which (51) holds ; and 

 if (51) holds for all values, k must be constant for all values 

 of «, /3, y. 



21. Going back to (29), taking Q great enough to allow 

 ^/Q to be neglected, and simplifying by (46), (43), and 

 (44), we find 



W .(I 2 , m 2 , rc 2 \f'< 2 ,„. 



and the problem (§ 17) of determining I, m, n, subject to (5) 

 and (6), to make l 2 /a. + m 2 /(3 + n 2 /y a maximum or minimum 

 for given values of X, ft, v, yields the equations 



eoX — co / l+ ~ =0 : cou — co'm + -^-=0 : cov — a>'n+ - = . (54), 

 a ' p 7 



to, 10' denoting indeterminate multipliers ; whence 



I 2 , m 2 , n 2 ,__ 



<°=; + J+y ^' 



a, 8 =P^ m '-^ + m !:L,'~h'+^U~l\ a (56), 



\ a 3 V/ 



'Z 2 1-m 2 

 ft)/* 



7 



7> 



7 





y . . . . (57). 



