Proceedings of Learned Societies. 77 



sa.2/= 1, as required by Jacobi's theory. It is not neces- 

 sary, I presume, to say anything about the second form of w. 



Mr. Cayley has not condescended to state his objections to 

 the reasoning in my last paper ; had he done so, I should have 

 had a chance of obviating them on his own ground, but as it 

 is I am left entirely in the dark respecting them ; and unhap- 

 pily the darkness is in no measure dispelled by the cloud of 

 mystery in which his last paper has enveloped the subject. 



I must now notice Mr. Cay ley's logic in the 16th No. of 

 the Cambridge Mathematical Journal. He was at liberty to 

 make u = w, or any other quantity; and in so doing he must 

 determine the true value of C, if he proceeded by a right 

 method. How then does it happen that for two forms of w he 

 obtains faulty or indeterminate results, no results as he calls 

 them ? Plainly because the forms of to were faulty, not from 

 any fault in the denominator. If Jacobi's denominator would 

 have set all right, how did it happen that he did not fall upon 

 it in these two cases? In the last case also he has in reality 

 arrived at no result; it was quite ridiculous to argue against 

 my denominator when he had obliterated it. If Jacobi's had 

 been the true form, he would in every case have fallen upon 

 it; and had there been no fault in the forms of w, he would in 

 no case have been led to faulty results. Had he carried his 

 method out fully, and drawn from it the proper inferences, he 

 would have proved all that I have asserted with regard to 

 Jacobi's functions. 



Gunthwaite Hall, Nov. 19, 1844. B. Bronwin. 



VII. Proceedings of Learned Societies. 



CAMBRIDGE PHILOSOPHICAL SOCIETY. 



Nov, 27, "/^^N the Foundation of Algebra," No. Ill, By Augustus 

 1843, ^^ De Morgan, of Trinity College, Professor of Mathe- 

 matics in University College, London, &c. 



In the second paper of this series a general definition of the ope- 

 ration A was laid down, A and B being each of them any form of 

 p-\-qA/ — \. The logarithm (or as Mr, De Morgan calls it, the logo- 

 meter^ of a line is thus described : — a line whose projection on the unit- 

 axis is the logarithm of the length, and whose projection on the per- 

 pendicular is the angle made with the unit-axis (or its arc to a radius 

 unity). Thus a line r inclined at an angle 6 has for its logometer a 

 line ^(\o^-r + d-) inclined at an angle whose tangent is 9 : log r 

 This being premised, the universal definition of A is the line whose 

 logometer is B xlogom. A. 



The object of this third paper is to show that the preceding defi- 



