50 Defence oj Mr. J. Herapath's Binomial Theorem. 



" It must be observed that when a difFei'ence of longitude 

 is expressed in time, the time intended is sidereal, and not 

 solar." 



If any difficulty should still remain after what has been 

 said, here is an authority that ought to have some weight. 



" La difference des meridiens en tems est toujours la dif- 

 ference du tems que Ton compte a un instant donne, soit que 

 les observateurs emploient le tems vrai, le tems moyen ou le 

 tems sideral, pourvu que les deux observateurs emploient le 

 meme tems." — Delambre's Asiron. vol. ii. p. 203. 



In the Requisite Tables there is a list of places with their 

 longitudes, reckoned from Greenwich, expressed both in de- 

 grees and in time. But I find no intimation that the time 

 meant is sidereal and not solar time. Yet, from the known 

 precision of the estimable author of the Tables, who presided 

 so long and with so much credit over the astronomical science 

 of this country, there is no doubt that such an intimation 

 would have been given, if it could have contributed in any 

 degree to remove misapprehension, or to guard against error. 

 If I have a clock regulated by sidereal time, and wish to know 

 the sidereal time at the meridian of any place in the Tables, I 

 apply the difference of longitude in time to the time of my 

 clock, and I have what I desire. If my clock is regulated by 

 mean time, I follow exactly the same procedure, with the like 

 success. In the first instance, the difference of longitude is 

 in sidereal time ; and in the second, in mean solar time. It 

 is always in that time according to which one reckons. 



July 18, 1825. DiS-lOTA. 



VII. Defence of Mr. J. Herapath's Demonstration of the 

 Binomial Theorem. By A Correspondent. 



To the Editor of the Philosophical Magazine and Journal. 



Sir, 



lY/TR. WARD has objected in your last to Mr. Herapath's 

 -'-*-*• demonstration of the binomial theorem, published in the 

 preceding Number. — It had been shown by Mr. H. that 



n— 1 n — 2 n — 3 ■,i—(m—\) ,■„, 



"' — ' — ' -T-' ••••— ^;r-^ (B) 



are respectively the quotients of the second by the first, the 

 third by the second, the fourth by the third, &c. to the (»j + 1 )th 

 by the viih coefficients of the expansion of {x + y)^; and hence 

 he immediately concluded that the ^^th coefficient is the pro- 

 duct of y — 1 of these terms, the coefficient of the first term 



being 



