194 ON THE CONSTANT TERM IN THE [25 



Hence, in the case supposed, the quantity --- cannot contain any 



T 1 T 



constant term of lower order than the third. 



More generally, the constant part of - - cannot be of a lower order 



than the constant part of the product of the quantity ^ multiplied by 



/*, r 



one or other of the quantities 



Now, as the two systems x, ij, z and x v y lt z l satisfy the same differ- 

 ential equations, the solutions can only differ from each other by involving 

 different values of the arbitrary constants. 



By applying the principle just stated to four different cases of variation 

 of the arbitrary constants, we shall be able to prove the properties already 

 enunciated, viz. 



E H . F 

 B = 0, (7=0, -, 



Let x = u cos (nt + e) r sin (nt + e), 



y u sin (nt + e) + r cos (nt + c) ; 



and similarly 



x, = ! cos (nt + e) i\ sin (nt + e), 



//! = w, sin (nt + e) + v 1 cos (nt + e), 

 where nt + e is supposed to retain the same value as before. 



Then (x - x$ + (y- ytf = (u- u,) 1 + (v- i\)\ 



Hence, in the statement of our principle, we may replace 



(x - xtf + (y- y^ + (z- zj'- -(r- r,)- 



by (t* - utf + (v - vtf + (z- zj* -(r- r$. 



For the sake of simplicity, we will take the quantity which was before 

 denoted by a as our unit of length, so that, instead of the quantity 



formerly designated by - , we shall write simply - . 



