OF DEFLECTIVE FORCES. 141 



the fluent of this equation, which in its present form cannot 

 be integrated for want of the relation between u and d^, 



we may multiply it by e ^ (cosyf + *^ — 1 sin yt) df, which 



we may call e^ Fd^, observing that dPziFv^ — \y(\t, and d 



mt 



m 



(e 2 T)zze 2 ^(9"+^ — ly)d^ Now, beginning with the 



first term — — , and taking the fluxion of its fluent multi- 



mt r ^ ddw *"' dw r ^ 



plied by e 2 r, we obtain^e 2 V——z=. e 2 f t~'j^ ^ ^ 



(— + '^—iy) du : the next step must, therefore, be with 



mt 



mdu—(^-^ n/ — ly)dM,or(- — ^/ — l7)dM ; and we have fe \ 



r (|_^^,)d«=e?r(f-N/=ir)u-/.?r (|-- 



V — 17)(-^ -f *^^y)dM : and this last term, that is^e 7 r 

 (— - + y2) u^ together with JcUy may be made to disappear 



by putting '^ +7^=it, and y= V(^ j-) = so that the 



whole equation will become e 2 (cos yf + v — 1 sm yt) f -j- 



+ — — >/— ly)MJ= — IfeT (cos yf + ^ —I sin yOw^df— ... 



If we compare the real and imaginary parts of this equa- 

 tion separately, which, as is well known, must always be 

 g5^ allowable, because imaginary quantities can never be 

 JP equated with real ones, unless they are compensated by 

 some other imaginary quantities, we shall obtain two equa- 



