272 
Proceedings of Royal Society of Edinburgh. [sess. 
Jacobi (Deer. 1831). 
[De transformations integrals duplicis indefiniti 
’ d<p dip 
A + B cos 0 + C sin 0 + (A' + B' cos 0 + C' sin 0) cos xp + ( A" + B" cos $ + C" sin 0) sin 0 
in formam simpliciorem J G _ G ' cos „ co7e-G".sin „ sin 6 ' 
— Crelle’s Journ ., viii. pp. 253-279, 321-357.] 
In his previous paper with a similar title to this Jacohi 
confined himself strictly to the consideration of his double 
integral, without saying a word as to the purely algebraical 
problem of transformation which lay at the root of it. Had he 
acted otherwise he would have been forced to note that this 
algebraical problem differed from that dealt with in the earlier 
paper of the same year merely in having four independent variables 
instead of three. Using modern phraseology, we may say that the 
one paper dealt explicitly with the transformation of a ternary 
quadric into the form L£ 2 + M.rj 2 + , and the other implicitly 
with the transformation of a quaternary quadric into the form 
G£ 0 2 + G'^ 2 + G"£ 2 2 + G'"£ 3 2 ; and such being the case, it is a matter 
for some surprise that the consideration of the corresponding 
problem for an n- ary quadric was left to Cauchy. 
In the lengthy paper we have now come to, the algebraical 
problem is no longer kept in the background, hut forms one of 
the three parts into which the subject-matter naturally divides 
itself. The first is the “ Introductio,” occupying §§ 1-9, pp. 253- 
264, and containing a brief account of previous related work, 
followed by an indication of the new results reached. The second 
des theoremes semblables aux siens, sans avoir connaissance de ses recherches. 
Le Memoire de M. Cauchy, et le mien, dont je donne plus loin un extrait, 
ont ete offerts le meme jour a l’Academie des Sciences.” A few pages further 
on in the same volume we come to an article entitled “ Extrait d’un Memoire 
sur l’integration d’un systeme d’equations differentielles lineaires, presente a 
l’Academie des Sciences le 27 Juillet 1829, par M. Sturm.” The abstract 
occupies nine pages (pp. 313-322), and though it does not contain explicit 
statement of the two theorems referred to by Cauchy, the theorems themselves 
are evidently implied. There can be little doubt, therefore, that the memoir 
here condensed is that which was presented on the same day as Cauchy’s. 
Probably the full memoir was never printed : Professor Gibson of Glasgow, 
who has kindly made a search, has failed to find trace of it. 
