580 



LAPLACE. 



Let ji stand for y^ ; then from (7) we can deduce the follow- 

 ing system of equations : 



1 = Xtaiji + fJ^afiiji + vtafiji + . . . ^ 



= XtaihJi+ fMtk^ji+ v%hiCiji-\- ... \^ (8). 



= Xtafiji + tJ&hiCiji + vtciji + . . . 



To obtain the first of equations (8) we multiply (7) by aji, 

 and then sum for all values of i paying regard to (6) ; to ob- 

 tain the second of equations (8) we multiply (7) by hji and sum ; 

 to obtain the third of equations (8) we multiply (7) by Ciji and 

 sum ; and so on. The number of equations (8) will thus be the 

 same as the number of conditions in (6), and therefore the same 

 as the number of arbitrary multipliers \ fju, v, ... Thus equations 

 (8) will determine X, fi, v, ... ; and then from (5) we have 



x = tyiqi + l (9). 



We shall now shew how this value of x may practically 

 be best calculated. 



Take s equations of which the type is 



GiX + hi7/' + CiZ + =qi + h. 



First multiply by aji and sum for all values of ^ ; then mul- 

 tiply by hji and sum ; then multiply by cj] and sum ; and so on : 

 thus we obtain the following system 



xta.^ji + ytafiiji + zt^a^Cij^ + . . . 

 xtapji 4- ythlj\ + zthiCiji^ + ... 

 x'tafiji + y^hiCiji + ztclji + . . . 



2 [qi -H h) aji 



^ {qi + h) hji 

 2 {qt + h;) c,j, 



r 



(10). 



Now we shall shew that if x be deduced from (10) we shall 

 have X = %%qi + I, and therefore x = x. 



For multiply equations (10) in order by X, ^i, v, ... and add; 

 then by (8) 



