596 
Proceedings of the Royal Society 
proportion, according as the direction of the motion of the ordinate 
is toward the + or - side of the absciss. 
When each fourth term of the series for d is taken, we obtain 
recurring functions of the fourth order; of these there are four 
varieties, distinguishable into two groups according as they involve 
odd or even powers of the primary. The curved lines representing 
the functions of the even powers accompany each other, crossing 
and recrossing on ordinates at equal intervals, the middle line be- 
tween them being a modification of the catenary. The lines repre- 
senting the functions with odd powers also accompany each other 
on ordinates midway between those of the previous pair. The dis- 
tance between these ordinates corresponds to the value of tt, the 
ratio of the circumference to the diameter of a circle ; and as the 
computation of this value is easily made, we have a new determina- 
tion of 7T, independent of the theory of the circle. The intersections 
of the curves of even with those of odd powers, are not on ordinates 
at equal distances. 
The quaternary functions are notable on this account, that by 
addition, they gave the catenarian — by subtraction, the circular 
functions. 
When we proceed to the fundamental recurring functions of 
higher orders, we find that the interruptions of the representative 
curves no longer occur on equidistant ordinates, although certain 
compounds of them present the plaited appearance of the ternary 
lines; and it is noteworthy, that then the loops widen as we pro- 
ceed towards the + end of the line of abscissse. 
3. The World as Governed by Law, Teleologically consi- 
dered. By K. S. Wyld. 
Mr Wyld stated that he considered the philosophic treatment 
of this subject important, as there existed a. great amount of loose, 
ill-digested opinion in the public mind regarding it, and possibly 
also in the minds of many men of science. 
The first object of the paper is to direct attention to the fact of 
the existence of general laws, alike in the physical and in the 
moral world ; to consider these as designed for the benefit of the 
