CERTAIN FACTORIAL EXPRESSIONS. 69 
ON CERTAIN RELATED FACTORIAL 
* EXPRESSIONS. 
By Evetyn G. Hoae, M.A. 
THE origin of the series of related factorial expressions 
given below may be stated as follows :—Let BC, the 
hypoteneuse of the right-angled triangle ABC, be 
divided in a point D so that the radu of the inscribed 
circles of the triangles ADB, ADC are equal and also 
let 
Ba ens re 
Then the following relation among the quantities involved 
may be easily proved, 
m2 (2b? — be) — 2be mn + (2c? — bc) n? = 0......(i) 
If now b = 2c, we get at once 
Te Ne 
If this value for the ratio m : n be now inserted in (1), 
the latter reduces to 
8b? — 25be + 18c? = 0. 
te. 2.27b? — (2 + 3)%bc + 2.3%ce* = 0 
(6 — 2c)(8b — 9c) = 0. 
The result of inserting in (i) the value 8b =9c gives 
7.8.n? — 2.8.9.mn + 9.10 m? = 0 
2.E. 2 (2n — 3m)(l4n — 15m) = 0 
By inserting in (i) the value 14n = 15m, we have 
2.14?.6? — (14 + 15) be + 2.15%.c? = 0 
(8b — 9c)(496 — 50c) = 0 
By continued insertion in (i) of the ratios 6: ¢ and 
m:n obtained by equating to zero the factors of the 
various resulting expressions in 0, c and m, n, the follow- 
ing series of identities may be obtained :— 
48. 49n? — 2.49. 50nm + 50. 51m? 
= 2(14n — 15m) (84n — 85m) 
2. 84752 — (84 + 85)* be + 85%c# 
= (49) — 50c) (2885 — 289e) 
~ 287. 288n* — 2. 288. 289nm + 289. 290m? 
: = 2 (84n — 85m) (492n — 493m) 
2.4927. b? — (492 + 493)? be + 2. 493.%c? 
= (288b — 289c) (16816 — 1682c) 
