8 REPORT — 1856. 



4. From observing the symmetry of formation 

 X(a'-')=tzi=-l 

 2(„P-'J'-')=tl_l£E!=+l 



^^a'-' t'-' c'- .0-) = ^ r!:r!-^"^ s + 1. 



and observing that 



p-\.p-^ />-r _^, 



1.2 r 



one cannot help guessing at the general theorem 



i;(aP-'6P-V-\... F"')=±l (mod p.). 

 according as the number of factors a,b,c,.. k is even or odd. 



But the process of determining the value of 2(0? 6? c*" A') in terms of 



the sums of powers of the roots is so laborious, that the law, which seems to 

 exist, has not been verified beyond four factors. 



The theorems might have been multiplied indefinitely ; but two only have 

 been selected, as being the most striking in their results. 

 5. Numerical examples : — 



1+2+3+4= 10 = (mod. 5) 

 P+2H3H4-= 30 = 



13 + 2H 3^+4'= 100 = 



14 + 2*+3*+4^= 354 = 4 



l8+2« + 3^ + 4^=72354 = 4 



1?2^+1?3'+1M' 

 + 2?3'+2?4^ + 3H'=2648I = l (mod. 5) 



1?2?3^+1?2?4^ 

 + i?3?4^ + 2?3H^=357904 = 4. 

 1?2?3M^=331776 = 1. 



On a New Method of Treating the Doctrine of Parallel Lines. 

 By Prof. Stevellt. 



The author stated that from the days of Euclid to the present, all geometricians 

 admitted that Euclid's twelfth axiom was a property to be proved, and not an axiom 

 to be assumed as self-evident ; but hitherto no satisfactory and sufficiently element- 

 arv proof of it had been adduced. He then showed that, by defining parallel lines 

 to be "when two lines in the same plane were both perpendicular to the same line, 

 they should be called parallel," all the properties of parallel fines as proved by 

 Euclid could be shown to belong to these, by two supplementary propositions. The 



