DIFFERENTIAL AND INTEGRAL CALCULUS. 55 



therefore 



3.*-* gjr-l x r 



x ( b -f- a 9 x + ...-}- -+...) = 0. 

 V [r-1 / 



whence, equating coefficients of af, 



tZ ~ P"^ ~ E~i = ' 



or a rfa = a r {r (r - 1) + r} = r*a r ; 

 therefore a z = ; therefore 4 = 0, 6 = 0, &c., 



and therefore 



la 3 1.3 a 5 1.3.5 a;' 



+ __ + _ _ + _____+. 



This is in fact y = a + l) sin" 1 ^, which satisfies the equation 

 equally well and is the general solution. 



DIFFERENTIAL CALCULUS. II. 

 1. Prove that 



dy dx dy dy dz dy dy dz 



dx dy ' dx dz dx du dz dx ' 



z being a function of x, and u of z. Find the differential 

 coefficients of 



cos 



* -O- ""' -O 



_JZx-x\ . 

 , tan ^^--3^] , log sins, log cosa?. 



