432 Proceedings of the Royal Society of Edinburgh. [Sess. 
noting the fact ; and (/3) is essentially the same as the second of the two 
results given in Hesse’s paper of 1847 (August), it being noted, however, that 
the case of this where r = s had been established by Hesse in 1844 (see 
Crelle's Journal , xxviii. p. 103, lines 1 and 2). 
Hesse, O. (1858). 
[Zur Theorie der ganzen homogenen Functionen. Crelle's Journal , lvi. 
pp. 263-269 : or WerJce, pp. 481-488.] 
The first part of Hesse’s attempted proof of his converse theorem of 
1851 (March) was to show that the vanishing of “the determinant of u” 
led to the establishment of a linear relation connecting the first differential 
coefficients of u. In this there was an oversight, which Brioschi repeated, 
but which Bellavitis and Baltzer, from the course followed by them, must 
have been conscious of. Accordingly, Hesse now returns to the subject, 
the one object of his six-page paper being “ diesen Lehrsatz strenger zu 
begrunden.” He first clears the ground a little by setting aside the case 
where one, and therefore all, of the primary minors of A vanish, merely 
stating that a linear substitution is then possible which will transform u 
into a function with two variables less than before. He then sets himself 
to supply the want which Bellavitis had drawn attention to, the result 
being a lengthy (pp. 265-268) and still unconvincing argument. 
LIST OF AUTHORS 
whose writings are herein dealt with. 
1844. Hesse 
1845. Cayley 
1847. Cayley 
1847. Hesse 
1849. Hesse 
1 849. Aronhold 
1849. Hesse and Jacobi 
1851. Terquem . 
1851. Hesse 
1851. Sylvester 
page 
413 
415 
417 
418 
418 
419 
419 
421 
422 
422 
1852. Salmon 
1852. Brioschi . 
1853. Sylvester 
1853. Spottiswoode 
1854. Brioschi . 
1854. Brioschi . 
1857. Bellavitis 
1857. Baltzer . 
1858. Hesse 
PAGE 
424 
425 
426 
426 
427 
429 
429 
431 
432 
(. Issued separately July 23, 1908.) 
