VOL. XX.] PHILOSOPHICAL TRANSACTIONS. 277 



' = & ^ = 0' / = ^' ^ = 0' ^^- h^"^^ ^ ^'^^ he = 711/ + ^V 

 &c, or ?2j/ + ■^-'^/'jy^i ^c. Supposing a to denote the whole preceding 



term, which will be the same series as Mr. Newton has first found. 



By the same method, this general problem may be solved ; the absciss cor- 

 responding to a certain area in any curve being given, to find the absciss, whose 

 corresponding area shall be to the first in a given ratio. 



The logarithmic series might also be found without borrowing any other idea, 

 than that logarithms are the indices of powers : let the number, whose logarithm 

 we inquire, be 1 + z, suppose its log. to be az + ^'zz + ca' &c. Let there be 

 another number \ + rj ; its logarithm will be ay + byij + cy\ &c. Now if 

 1 + z = 1 + ?/!", it follows, that az + bzz + cz' &c : ay + byy -\- cy^ 

 &c :: n : 1, that is, az + Z'zz + cz' &c. = nay + nhjy + ncy^ &c. There- 

 fore we may find a value of z expressed by the powers of j/ ; again, since 1 + z 



= 1 + y\\ therefore z = 1 + y^" — 1, that is z = ny + y + ^^T~^'^ "^ 



^ X ^— X ^— -^ ?/' &c. Therefore z is doubly expressed by the powers of y. 



Compare these two values together, and the coefficients a, b, c, &c. will be de- 

 termined, except the first a which may be taken at pleasure, and gives accord- 

 ingly all the different species of logarithms. 



Account of an extraordinary Iris, or Rainbow, seen at Chester. By E. Halley. 



N° 240, p. 193. 



August 6, 1698, in the evening, between 6 and 7 o'clock, walking on the 

 walls of Chester, I was surprised by a sudden shower, which forced me to take 

 shelter in a nich in the wall, I soon observed an iris, exceedingly vivid, as to 

 its colours, at first on the south side only, but in a little time with an entire 

 arch ; and soon after, the beams of the sun being very strong, there appeared a 

 secondary iris, whose colours were more than ordinary bright ; but inverted, 

 as usually : that is, the red was inwards, which in the primary iris is outward, 

 and e contra for the blues. But what appeared most remarkable was, that with 

 these two concentric arches, there appeared a third arch, nearly as bright as the 

 secondary iris, but coloured in the order of the primary, which took its rise from 

 the intersection of the horizon and primary iris, and went across the space be- 

 tween the two, and intersected the secondary, as in fig. 6, pi. 5, apcg inter- 

 sects the secondary iris efgd, dividing the arch ed nearly into three equal parts : 

 but at first the arch af did not appear, which afterwards became as brighi as the 

 former. I observed the points f and g to arise, and the arch fg gradually to 



