SIR r. POLLOCK OK FEEMAT’S THEOEEM OF THE POLYGONAL NUMBEES. 415 
n, n, n, w+1; but every term will be composed of one or more square numbers + 
arithmetic number, and the squares and the arithmetic numbers will each form a regular 
series. An example in figures ufill best explain this : 1 9 =:_9 + 10 = 4 + 6 + 5 + 5. If 19 
2 
be increased by 2, 4, 6, 8, &c., the 9th term is 91 = 4, 5, 5, 5 ; so if 19 be increased by 
2, 4, 6, 8, &c., the successive terms udll be composed of squares and arithmetic numbers 
as below : to distinguish the arithmetic numbers from roots, I enclose them in a ( ) . 
i 
xS umbers. 
Eoots. 
Numbers. 
Eoots. 
19= 
0,1 
or = 
© 
1,1 
21 = 
® 
1,1 
= 
1,2 
25 = 
(§) 
1,2 
© 
2,2 
31 = 
@ 
2,2 
z= 
® 
2,3 
39 = 
(g) 
2,3 
— 
@ 
3,3 
49= 
@ 
3,3 
= 
® 
3,4 
61 = 
(§) 
3,4 
® 
4,4 
75 = 
4,4 
© 
4,5 
91 = 
@ 
4,5 
= 
© 
5,5 
1 
1 
(or 5, 5) 
109 = 
! 
(59) 
5,5 
= 
® 
5, 6 
(See. 
&c. 
&C. 
&c. 
