112 Biographical Notice of [Feb. 



-s 1 = ( V — 1) °, the t-^ — : power will be equal to the first 



simply, or to «/ — 1 ; and the -^- — { to the — first, or to 

 — */ — 1 . Expanded into series, these values are 



C — +' 2 m + n 1 T" + I . 304 m" + 2I0mn + 35n« 



»»±- {*"+«*■■ +-^r a * x + *Toi? 



— +1 2064 jw3 + 1636m 5 rc + 420 m n* + 35 tfl . ~ + * , Qr ? 



a 3 x » j __ . a 4 x + &c. r 



C — + » 2m+» — +« 304 m* + 210 m n + 35 u» 



*~± [*-«*' +-^T fltr " sio? 



6m 8 m ~\ 



. — + ' 2064 »i3 + 1636 m?n + 420m n- + 35 «3 — + i / 



«3 * » j . a* z — &c. c 



* T 840 »3 J) 



Ex. 3. — For instance, if m = 3, ?? = 1, and r/ = 1, then are 



z = - {x + x 7 + * x 13 + y T V ^' 9 + 7 ^- 7 * 25 + &c.} 



a; = _ { Z - tf + 7. -13 _ S^jM z .9 + 7£J4 7 -« _ &c . } 



An endless variety of series endued with the same property 

 may be educed from the general formula?, by making <p x = a 



sine, tangent, cosecant, on their arcs, to log. t— — , or in short to 



any function containing only odd powers of x. Any other would 

 be encumbered with imaginary coefficients. 



Article VII. 

 Biographical Notice of the Rev. W. Gregor. 



The Rev. William Gregor, descended from a line of ances- 

 tors who have held a distinguished station in the comity of 

 Cornwall for a long period of years, was born in the year 1762. 

 He received his school education, at Bristol, under the Rev. Mr. 

 Lee, an able classical scholar ; was for a short time under private 

 tuition at Walthamstow, in Essex ; and at the usual age entered 

 a pensioner at St. John's College, Cambridge. He proceeded 

 A. B. in 1783, and A. M. in 1786, and was elected Piatt Fellow 

 of that Society. His classical and mathematical attainments 

 were of a high order, and procured for him the most distin- 

 guished honours of the university- 



In 1789 he was collated to the rectory of Deptford, near 

 Totness, upon the presentation of his father; and in the succeed- 

 ing year he married the daughter of Edward Gwatkin, Esq. In 

 1793 he was presented by Bishop Ross to the vicarage of Brat- 

 ton Clovelly, in Devon, which he soon after exchanged for the 

 rectory of Creed, near Grampound, in the county of Cornwall. 



