210 



SCIENCE PROGRESS. 



x 2 y 2 z 2 q, 2 q 2 , . . . etc., and also of the products q T q 2 , . . . 

 etc., by integrating the expressions — 



£ -k(mi* + my* + mi* + a 1 q\»+b 1!l g\q\ + . . ■) X ^X d)' d'z dq ,, . . . 



etc., between the limits + co for each variable. The result 

 is as follows : — 



Form the determinant 

 2m 



2771 



2771 



2a, b l2 . . 

 b 12 2Cl z . . 



d 





2a 



» — 3 



And let d lt d l2 d 22 , etc., be its minors. Then we find 



""■ ~" ■" U,t W"m. &\\ 



n i 1 JJ 



x ~ = y - z = id m id 



g * ~ hd 



d 



45 



• ?' 9* : U 



, etc., 



and therefore m(x ' 2 + y 2 + z 2 ) = ~r> 



also a, q 2 + 6 ia q, q 2 + etc., = 



« - 3 



2 k 



Tl f (^ 2 + / + 2 2 ) 



1 herefore w — — ^ J — '- — 



a* q? + K % q 1 q 2 + etc. 



or 77i (^ + f + z 2 ) 



n 



2 T 



71 



It is not generally true that a x q 2 = a 2 q 2 , etc. But it is 

 possible, for given values of the co-ordinates, by a linear 

 transformation to reduce the quadratic function to the 

 fonruv/7 + 2 / 2 2 » etc., containing no products //, and these 

 quantities q have the property that a 2 q 2 = a 2 q 2 = , etc., if 

 in this form they are of any use. The proof that 



i7i (x 3 + f +.2) = 3 ^ oes nQt depend on t h e transformation. 



2T 71 



15. It follows from this theorem, as shown first by Max- 

 well, and as explained in Watson's book, that for a gas 



