Thsrefore we may find aValue oiz, expreft by the Powers of 
y ; again^ fince i-^- /\ therefore i -j - 
n n — I n r-^i n — 2 - 
that is ny^ - X — ;^ + x — x ~ y^^ 
&c. Therefore is doubly exp'-tft by the Powers of /. 
Compare thefe Two Values toge her, and the Co efficients 
ay by &c, will be determined, except the firft a which may 
be taken at Pleafure^ and gives accordingly^ all the different 
Species of Logarithms. 
XL An Accomt of the Appearance of an ex- 
traordinary Iris feen at Chcfter, in Augufl: 
laftj by E. Halley. 
ON the Sixth Day of Juguji laft, in the Evening, 
between Six and Seven of the Clock, I went to 
take the Air upon the Walls of Chefier, when I was fur- 
prized by a fuddtn Shower, which forced me to take 
Shelter in a Nich that afforded me a Seat in the Wall, 
near the North Eaft Corner thereof. As I fat there, I 
obferved an Ir^, exceedingly vivid, as to its Colours, 
at firft on the South Side only, but in a little Time with 
an entire Arch ; and foon after, the Beams of the Sun 
being very ftrong,there appeared a fecondary /r/j,who(e 
Colours were more than ordinary Bright but inverted, 
as ufually : that is, the Red was inwards, which in the 
primary Iris is outward, and e contra for the Blues, 
But what I took moft Notice of was, that with thefe 
Two concentrick Arches, there appeared a Third Arch, 
near upon as bright as the Secondary Iris, but coloured in 
the Order of the Primary, which took its Rife from the 
Interfedion of the Horizon and Primary Irij, and went 
F f ^ crofs 
