AND THE NEBULAR HYPOTHESIS. 473 



Then the solution (48) may be replaced by 



(50) 



which is formally analogous to (7). 



The following properties of the function J. + .,,(a-, 6) may readily U< verified, an.l 

 will be required later : 



. 



(52) 

 (5:1) 



There is a ready rule for writing down the values of these functions. In the first 

 place, we have 



T , , sinx cos x 



J /,. W = /T . J -v, W = 

 V^TJ- 



so that 



. sin(.r + 0) 



J,,,*, = - 



V ^Tr./' 



Now let ^(.e + fl) be used to denote a general function made up of circular functioiiH 

 of x + 6 and of algebraic functions of x. Then J>/,(- r > 6) is of the form <f>(j: + 6), and 

 any number of differentiations with respect to x, or of multiplications by powers of x, 

 will still leave it in this form. It follows from (52) that !/,('', 0), Jv,(ar, 0), Ac., 

 will l)e of this form. Hence we have the general law 



< 54 ) 



in which the functional form of $ is at once given by 



#(*) = !.+'/, (-4 



For instance, 



/o \ 3 



J,, ( x ) = Pj - 1 ) sin x- - cos x, 



\3t / X 



so that 



J.,,(ir, 0) = *(* + <>) = (^-l)sin(a- + 0)-|co8(z+0). | 



17. We proceed to carry out analysis similar U> that of 13. Suppose that under 

 a tide-generating potential r n S n , and a rotation w, the core assumes a configuration 

 such that its boundary r = <t becomes deformed into 



. . I 55 ) 



VOL. (10X111. A, 



