J. B. Luce on the Theory of Numbers. 55 



ness, and there are often several layers at different elevations, 

 they will be frequently projected upon each other. Or the case 

 may be stated thus ; if we suppose isolated clouds to be scattered 

 uniformly throughout the cloudy stratum, the line of sight which 

 makes a small angle with the plane of the horizon will cross this 

 stratum very obliquely, and be proportionally more likely to meet 

 with a cloud than a line traversing this stratum perpendicularly 

 at the zenith. Besides from the obliquity of this line a haziness 

 which would scarcely be observed in the zenith might amount 

 to a positive cloud at a distance from the zenith. This is well 

 illustrated when a fog or mist is gradually being dissipated, and 

 the sky is first seen in the zenith. 



In this way it may very well happen that, although there may 

 be considerable differences among the numbers representing the 

 cloudiness of the sky in general, for the quarters of years, there 

 may be no material variations in the nocturnal decrements of tem- 

 perature, corresponding to these differences. 



(To be continued.) 



Art. V. — On the Theory of Numbers ; by J. B. Luce. 



Having read with much satisfaction, in the American Journal 

 of Science and Arts, the very ingenious method of interpolations, 

 as explained in a plain, elegant and very satisfactory manner by 

 J- H. Alexander ; it induces me to send you my new theorem 

 in the theory of numbers. I call it new, because it is probably 

 so, having never read it in any treatise on the theory of numbers, 

 and particularly in that of Peter Barlow, who, as it is well known, 

 published a useful and elegant book on these matters. 



The equation^ 2 -nq 2 = \ is well known to be a fundamental 

 one in the resolution of indeterminate problems of the second 

 degree, since, in frequent cases, this solution depends on the find- 

 ing of integral values for p and q. I use here the same symbols 

 as Peter Barlow. But not having now with me his book, I do 

 not recollect precisely the method he uses to find integral values 

 for;? and q; all I remember is that it is performed by a long and 

 tedious process of continued extractions of the square root of ^^, 

 coefficient of q\ a method which I laid aside since the finding 

 °f the theorem which I am going to expose, and may, in all 

 circumstances, supply the old method. 



I propose to myself the resolution of the more general equa- 

 tion x* - wy3 = z > where i may be 0, 1, 2, 3, &c. It is evident 



that we can always assume rc^a 2 ±6. Hence v/n = v/ a 2 ±i. 



Assume ^a*±b=a±f which gives/ 2 ±2af= ±6; which ex- 

 panded in a continued fraction, and, for abbreviation's sake, putting 



