LAGRANGE'S AND LAPLACE'S THEOREMS. 12] 



which becomes 



^ (- 1)"' 1 (1 + nf l e cos fa + z - (n - 1) </,}, 



where cot <p = n, by a process similar to that in Art. 81. 



Putting z = in this, "we have for the required expansion 

 sin (a + y) = sin a + x cos a + . . . 



+ (- I)"' 1 (1 + rtfr cos (a - (n - 1) cot' 1 w} + "... 



6. Given ay + x logy = 0, find sin y in powers of x. 



7. Given y z + xy p e w , expand y'V" in powers of x. 



8. Given y = z + x siny, expand sin y and sin 2y in powers 

 of #. 



9. Given y = log (z + x cos y), expand e v in powers of x. 



10. From the equation xy* + 2xy 5 + Zxy* + 2y + 1 = de- 

 termine y in ascending powers of x. 



19 9 1395 , 

 y= ~2 "82 *-32* -4096*- 

 * 



11. If ye* ' 8V , find the first four terms of the 

 expansion of cos log y in powers of x. 



1 x 



- 



s 



12. If y 8 + my*+ ny = x, shew that one value of y is 



x m (x\* 2m?nfx\ 3 5m 3 5mnfx\* 

 n 



x m /x\ 2 2m? n Sx\ 3 5m 3 5mn /x\* 

 n n \nj i? \nj n 3 \nj 



