E s s J; 
V. An Appendix to the Paper in the Philof optical Pranf- 
aElions for the Year 1778, Number xlh, pages go 2 
et feq. intitled , “ A Method of extending Cardan’s 
“ Rule for refolving one Cafe of the Cubick equation 
“ x l -qx~r to the other Cafe of the fame Equation, 
u which it is not naturally fitted to folve , and which is 
a therefore called the irreducible Cafe.” By' Francis^ 
Maferes, Efq. F. R. S. Curjitor Baron of the Exchequer „ 
Read Nov. 4, 1*779;. 
Article I. 
I N the above-mentioned paper in the Philofophicaf 
Tranfadlions the expreffion \/px the infinite feries 
2+— - -^L + - &c. is fhewn to be equal to the 
root of the equation x 3 - qx =r, whenever ~ is lefs than 
3 3 
E y but greater than one half of it, or than q —. This 
expreffion is wholly tranfcendental, or compofed of an 
infinite number of terms, to wit, the terms of the fe- 
ries a-+— — -FfL + - 8cc. multiplied into the cube- 
root of e. But I have fince thought that it might be con- 
venient on fome occafions to divide this expreffion, if 
poffible, into two others, whereof the one ffiould be a 
3 , 
mere 
