Mr. J. Cockle on Transcendental and Algebraic Solution. 137 



Next, to determine these arbitrary constants. Multiply (1) 

 into 4, and in the product substitute for x its value given bv (3). 

 Then, eliminating the cube of the sine by the known formula of 

 trigonometry, the result is 



3(A 3 -4A)sin( Sm ~— + b) 



-A 3 sin(sin- 1 « + 3B) + 8« = 7 

 which is satisfied if 



A 3 -4A=0, A 3 =8, cos3B = l, sin3B = 0; 



that is, r being an integer, if 



A = 2, 3B = 2r<7r. 



Hence _ . /sin _1 «-f 2riT 



# = 2 sin I - 



: )- 



(« 2 - 1 )^+«^-^ = ^ W 



\ 3 



This discussion embraces the " irreducible case ;" but if a be 

 greater than 1 we must employ logarithmic in place of trigono- 

 metric forms. Putting (2) under the form 



d^x dx 

 da 2 da 



where m — \ y we find that (4) is* equivalent to the symbolical 

 equation 



whence f 



order be divided by the last, and the square root of any multiple of the 

 quotient, be integrated, the form of the integral occasionally suggests a 

 convenient transformation. Thus, for (2) let 



J da 



then, a being determinable as a function of t (for a= sin t), if t be made 

 the independent variable, we see a priori that the first and last coefficients 

 of the transformed equation will be constant. In the present case, indeed, 

 all three are constant. 

 * In this case 



r da 



Clog(a±Va 2 -l); 



and, as before, a is determinable as a function of t, and all the coefficients 

 of the transformed equation are constants, the middle one vanishing, 

 t In the equation 



W is always determinable as a function of x. For 



, dW 



is a linear differential equation. 



/ dW , w 

 yx . — — + xx . W = px 

 ax 



