Mr. Walsh's Account of the Binomial Calculus. 443 



For calculating the differences of right ascension n and 

 e are sufficient; from the 27th May 1820, only n tgl+c sec c 

 was added to the observations under the head Corrections des 

 Instruments, where c as immediately found was corrected by 

 — 0"0121 on account of the daily aberration. If, therefore, 

 after that time the times of passage over the true meridian 

 are required, m is to be added, which is deduced from the for- 

 mula m = n. cot. g <p-\-a cosec. p, where 1 5a is supposed to be 

 the eastern azimuth of the instrument observed by the me- 

 ridian marks and corrected for its deviation, for the beginning 

 and the end of the period. The double determinations of m 

 hence resulting, and registered in the last column of the Journal, 

 belong therefore to the limits of the period f. e. from June 12 

 —27 m=+0"-20 and 0"-51 ; the difference is involved in the 

 rate of the clock, and has therefore no further influence. 

 [To be continued.] 



LXXII. Some Account of the Binomial Calculus. By 

 J. Walsh, Esq. 



TN the year 1815, having met with an accident which for 

 -* some time confined me, I resumed the study of mathema- 

 tics, of which I had before some knowledge of the elementary 

 branches. The figure of the earth and the precession of the 

 equinoxes first excited my attention. I did not believe the 

 received theory of the earth ; nor that the other proceeded 

 from the action of the sun on the excess of matter at the 

 equator : because, granting even that the sun possessed an at- 

 tractive force, there could be no accumulation of matter at the 

 equator, but an extension of the parts arising from the centri- 

 fugal force of the earth. If the equatorial parts became more 

 extended, the polar parts became more condensed; the quantity 

 of matter remaining the same in every section taken in any di- 

 rection from the centre of the earth. Such were my opinions 

 then : nor are they yet altered ; but I was totally inadequate 

 to investigate them, as I had no knowledge whatever of the 

 higher mathematics. I gave up then any further thoughts 

 npon those subjects. But to prepare myself for investigating 

 them at some future time,— and with this intention only, — I 

 commenced the study of the infinitesimal analysis, as it was 

 called. I got some introductory works on this subject; but 

 as I could not understand the principles on which they pro- 

 ceeded, I abandoned them as soon as I got them. I gave up 

 then entirely the idea of prosecuting such a study any longer; 

 especially, when I demonstrated that it was grounded on ab- 

 3 K 2 surd 



