430 Mr. J. J. Sylvester on Multiplications, ^c. 



my priority in the discovery of the remarkable theorems which 

 I had succeeded iu demonstrating. After I had signified my 

 withdrawal, one of the reporters submitted to me of his own 

 accord the reasons which induced him to form an unfavourable 

 opmiou of the paper, in perfect confidence that these reasons 

 would convince me of my errors. The arguments are mainly 

 directed against the truth of the new equation 



(see Phil. Mag. for April, p. 280), which is the principal feature 

 of my paper, and with which the theorems stand or fall. Now 

 what distinctly marks the character of these arguments is, that 

 thei/ contain not the slightest allusion to the reasoning by which the 

 equation is established. They are all drawn from extraneous con- 

 siderations, or from supposed consequences of the equation, and are 

 all such as I would readily answer if they were advanced publicly. 



In fact, the proof of the equation is too simple and straight- 

 forward to admit of any question. In support of this assertion, 

 I appeal to the reasoning given in my communication to the Phi- 

 losophical Magazine ; and I may add, that in an examination 

 for the award of Dr. Smithes prizes held in January last, I pro- 

 posed this equation for the candidates to investigate, when it 

 was impossible they could have known of its existence till the 

 examination paper was placed before them, and I received a com- 

 plete proof of it. This circumstance sufiices to show that it 

 requires for its demonstration nothing but mathematical prin- 

 ciples and processes commonly received and taught. 



I now invite the reporter on my paper to discuss with me in 

 this Journal (anonymously if he pleases), the important and in- 

 teresting questions in the Lunar Theory raised by my communi- 

 cation to the April Number. 



I am. Gentlemen, 



Your obedient Servant, 



Cambridge Observatory, J. Challis. 



May 22, 1854. 



LXXI. Note on a Formula by aid of ivhich and of a table of 

 single entry the continued product of any set of numbers {or at 

 least a given constant multiple thereof) may be effected by addi- 

 tions and subtractions only without the use of Logarithms. By 

 J. J. Sylvester, M.A., F.R.S."^ 



Introduction. 



THE remark to which this note refers is not new ; it has 

 been well observed somewhere in Gergonne's Annates 

 (Mr. Cayley being my informant), that by aid of the formula 

 * Communicated by the Author. 



