230 Proceedings of Royal Society of Edinburgh. [ sess . 
JACOBI (1831-33). 
[De transformatione integralis duplicis indefiniti 
f dcfidxf/ 
A + Bcos <£ + C sin <f> + ( A' + B' cos </> + C' sin </>) cos if/ + (A" + B" cos $ + C" sin<£) sin ^ 
in formam simpliciorem Z' — • : — . 
G- G cos^costf- G sm^sin# 
[ Crelle’s Journal , viii. pp. 253-279, 321-357.] 
[De transformatione et determinatione integralium duplicium com- 
mentatio tertia. Crelle’s Journal , x. pp. 101-128.] 
[De binis qnibuslibet functionibus bomogeneis secundi ordinis per 
substitutiones lineares in alias binas transformandis, quae solis 
quadratis variabilium constant ; una cum variis theorematis de 
transformatione et determinatione integralium multiplicium. 
Crelle’s Journal , xii. pp. 1-69]. 
Tbe first two of these memoirs may be viewed as continuations of a 
memoir with a similar title, which appeared in the second volume of 
Crelle’s Journal , and to which we have already referred. They are 
noted here merely in order that the thread of investigation may be 
preserved unbroken, for the last memoir practically swallows up, by 
means of its splendid generalisations, all those that had gone before. 
So long as we confine ourselves, in problems of transformation, to 
three independent variables, the explicit employment of the theory of 
determinants may be dispensed with. When, however, a sufficient 
number of special cases have been investigated, and an alluring 
glimpse has thereby been got of a generalisation involving them all, 
he who attempts the establishment of the generalisation must have 
recourse to the new weapon. In this latter position Jacobi now 
found himself. He wished to pass from the problem of orthogonal 
substitution in the case of three variables to the analogous problem 
in which the number of variables is ?i, or in his own words (p. 7) : — 
“ Investigare substitutiones lineares huiusmodi 
y ^ = 4* 0*2 4* • • • • “H CL n 5 
y 2 — cq" x x + a 2 " x 2 + . . . . + a w " X n , 
y* 
a^ n) X x + a 2 {n) X 2 + .... + a n [n) X n , 
