1 34- Mr. J. Cockle's Solution of two Geometrical Problems. 



distant from B and D, AD = AB. Let AD =2;; then, we 

 may express the two last conditions by 



z = a-\-Xi ....... (6.) 



and 



z=ai (7.) 



add (6.) and (7.) and we obtain, on reducing, 



z=«+|, ....... (8.) 



or 



^ T ^ " ^ -^ 



Now, in its present form, the last equation does not furnish us 

 with any solution of the problem. But it may be rendered 

 available by decomposing it into two congeneric surd equa- 

 tions and selecting the impossible congener. For, the left-hand 

 side of (9.) may be resolved into two factors, both of which 

 are included in the expression 



±^Z + ^^---^-^+Z. . . (10.) 

 Assume that 



""-^-'-^ +Z==;Z, . . . (1.1.) 

 then one of the values of (10.) takes the form 



and, consequently, vanishes. This, then, is the solution of 

 (9.) which we are in quest of, and, further, this is a solution 

 which must not be neglected supposing that we admit impos- 

 sible quantities into algebra. We must now consider D as 

 situate out of the plane of ABC. 



But, the question occurs, which of the infinite number of 

 values of Z are we to select ? The answer is, that which sa- 

 tisfies the condition* that the orthographical projection of D 

 on the plane of ABC shall be the centre of the circle inscribed 

 in ABC. But, this condition gives, 



whence, 



r, Sa 



* The condition in question is, perhaps, the one best adapted to the 

 problem before us ; the determination of Z under its most general aspect 

 will be discussed on a fitting occasion. 



