92 DIFFERENTIAL AND INTEGRAL CALCULUS. 



and the final equation is 



whence z = A cosnO + B< sin?i$, so that the complete solution 

 of the original equation is 



y = (l + a? 2 )~* n {A cos (rc tan" 1 ^) -f B sin (n tan" 1 a;)}. 



Differential equations involving #-T^J x * ~j * are mucn 

 simplified by taking x *, for 



dy _dy ^ dx _\ dy dy _ dy . 



~T~ ~T~ ~ ~~T~ ~T~ i &.C. t*/ j 7 . 



c?o; c?2 c?2 a; a^ 7 ax dz 7 



therefore x ^ + -^ = -A ~ 5 



dk' dz dz* x j 



>,d*y d*y dy did \ 

 therefore a; T4 = 7 ? -- r- = i fr 1 1 V t 

 ^ dz' dz dz \dz J J ' 



.,d 3 y d*y 1 d / d \dy 

 therefore ^ J + 2a; _| =--(-- ,)-|, 





.(Fy (d \ d? (d \ d 



* -7-3 = \T - ! -j-* y - 2 I -T- - 1 J -r V 

 dz 3 \dz J dz zy , V& /^^ 



and so on, the general formula being 

 n d n d d \ d 



GEOMETRICAL APPLICATIONS. 



If x, y be any point P on a curve, x -f Aic, y + A?/ a 

 contiguous point $, (A 7 ", 1 T ) coordinates of any point R on 

 the straight line PQ, then, as in fig. 19, 



-nQ Y-y _ A /, 



Pn ' X-^~ A* 



