of Edinburgh, Session 1880-81. 173 
which means that a child of the father of any woman B cannot be 
1 
the child of a daughter of the father of B. Hence %c m - 'c f B = 0, 
C 
that is, a child of the father of any woman B cannot be the child 
of B. And this equation is equivalent to the preceding, because 
f B may be any woman. Again, it follows from the original 
equation that - c”y- = 0, that is, a mother of a child of the 
c c 
man A cannot be a mother of a child of a daughter of the man, 
hence 
2 / “ e m yc m A = 0 . 
0 
It is evident from § 18 that an equation of this kind can be trans- 
formed by operating both in front and at the end of a factor, and hence 
the following rule To transform a universal equation which has a 
compound term of the second degree equated to 0, suppose all the 
symbols brought to one factor in accordance with the Ride of % 17, 
then removing a symbol from the front gives one derived equation , 
and removing a symbol from the end gives another derived equation. 
Transform each of these two in a similar manner , then each of their 
four resultants , and so on until all the terms have been brought to 
the other factor. The total of these derived equations is the total 
number of transformations of the given universal equation. 
5. Note on a Singular Problem in Kinetics. 
By Professor Tait. 
The following problem presented itself to me nearly thirty years 
ago. I cannot find any notice of it in books, though it must have 
occurred to every one who has studied the oscillations of a 
balance : — 
Two equal masses are attached to the ends of a cord passing over 
a smooth pulley (as in Attwood’s machine). One of them is slightly 
disturbed, in a vertical plane , from its position of equilibrium. Find 
the nature of the subsequent motion of the system. 
The interest of this case of small motions is twofold. Prom the 
peculiar form of the equations of motion, it is of exceptional mathe- 
matical difficulty. This is probably the reason for its not having 
