59] CALCULATING THE TRAJECTORIES OF SHOT. 475 



and the last with 



1 1 n , f" 1 du . k(n-l) f a 



-- = -* 



This may be done by means of the following lemma, which follows 

 immediately from Taylor's theorem : 



Lemma. 



If F(<f>) be any function either of <f> only, or of <f> and u, where u 

 is a function of <f> given by the above differential equation (l), and if a 



and ft be the limiting values of < in the integral and y = - (a + ft), then, 



Zi 



putting for a moment <f> = y + (a, 



F(<f>)d<j>= 



ft -i(-/3) 



+ F " + F> " + F "" + &c 



,,, . . 



where F'(<f>) = - , *(*)- - , &c., 



and ^(7), ^'(y), -F"(y), &c., are what J P(^>), J F / (<^.), J P // (<^)), &c., become when 

 y is substituted for <, and the corresponding value of u (u suppose) is 

 put for u. 



In what follows, the last of the terms above written, which is of the 

 5th order in (a ft), is neglected, together with all terms of the same 

 order of small quantities. 



All the definite integrals with which we are here concerned are included 

 in the two forms 



fa. fa. 



u 1 (sec (j)) m d(f>, and u 1 (sec tf>) m tan < d<f>. 

 Jft J P 



In the first place, we will apply the above formula to the case in 

 which F (<) is a function of < only, viz. when F (<) = (sec <)" +1 . 



602 



