14$ Mr. Gompertz on Series which may be summed 
cular cases in which both Mr. Landen's means and my own 
fail : I have added towards the end a general scholium con- 
cerning the cause, circumstances, and consequences of such 
failure in my method. 
The foundation of the theorems depends on the following 
well known lemmas. 
No. I. 
2 sine of vz . sine of tz, is equal to 
cos. of t — v . z — cos. of t -j- v . z. 
No. II. 
2 sine of vz . cos. of tz, is equal to 
sine of t -J- v . z — sine of t — v . z, or 
sine of t + v . z -f- sine of i > —t .z. 
No. III. 
2 cos. of vz . cos. of tz, is equal to 
cos. of t — - v .z -f cos. of t + v . z. 
Theorem I. 
If there be an infinite series a . sine oipz,-\-b . sine of p-\-q . z, 
+ c . sine of p-\-zq . z,\d . sine of p-f-sq • z & c - = 
and from the series a, b, c, d, e, f, &c. there be conti- 
nually formed new fa, a', b', c', d', e' , &c." 
series a, a!' , b" , c", d", e", &c. 
1 a,a , ",b" , ,c'",d , ",e'",&LC. 
_&c. &c. &c. &c. &c. 3c c. &c._ 
Every new series, being formed from that immediately 
above, by taking the differences of the terms, exactly in 
