168 Proceedings of Royal Society of Edinburgh. 
[sess. 
Theorem regarding the Orthogonal Transformation 
of a Quadric. By Thomas Muir, LL.D. 
(MS. received July 27, 1903. Read November 2, 1903.) 
(1) The theorem in question arises out of a consideration of 
several passages in Jacobi’s important memoir of 1833 * on 
orthogonal transformation. Having determined the substitution 
which simultaneously changes 
x 1 2 + x 2 2 + . . . + x 2 into yf + y 2 2 + . . . +y n 2 
and 
into G Y y 2 + G 2 y 2 + . . . + G n y n 2 , 
Jacobi proceeds to show (p. 12) that, by the same substitution, 
Gfy 2 + Gfy 2 + . . . + G *y H * 
where p is any positive integer, can be expressed in terms of 
x Y , x 2 , . . . , x n (“ expressionen per ipsas x l , x 2 , . . . , x n exhibere 
licet”). The actual result, however, is not sought for. Later on 
(p. 14) he reaches a theorem which would enable him to remove 
the restriction on p so as to admit negative integral values as well, 
but the opportunity is not used. The reason for the seeming 
neglect probably is that he has in view a second return to the 
subject when prepared to deal more effectively with it. However 
this may be, certain it is that he does return to it, and gives a 
hypothetical form of the desired expression in x 1 , x 2 , . . . , x n . 
His words (p. 20) are : — 
“ Statuamus Gfy 2 + G Sy 2 2 + . . . +G py 2 = nbi 
/cA 
+ Gf<l 2K (l 2 ^ + . . . CL nK CL n \ — Pk\i 
and where, we may add, the a’s are the coefficients of the substitu- 
tion. Regarding the validity of this nothing is said, but proof is 
* Jacobi, C. G. J., De binis quibuslibet functionibus homogeneis secundi 
ordinis per substitution es lineares in alias binas transformandis 
Crelle’s Journ., xii. pp. 1-69. (Aug. 1833.) 
