152 
Proceedings of the Poyal Society 
rivatives, thus bringing the problem under the dominion of the 
calculus of primaries ; the object of that calculus being to discover 
the relation of the primary variable to the function, when the 
relation subsisting between that function and its derivative is 
known. It also shows that the function belongs to the class of 
recurring functions of the fourth order ; and that, therefore, the 
equation 
<Px = y = A 0 j + B05 + C0? + D0f 
represents the form of the curve in every possible case of simple 
vibration ; A, B, C, D being coefficients, having fixed ratios to 
each other, and l being a length depending on the relation of the 
linear unit to the dimensions of the system. 
It onty remains to apply this general formula to particular cases. 
The end of an elastic bar may be held firmly as in a vice, it may 
be entirely free, and it may be touched, but without any angular 
tension ; and these three conditions may be combined at the two 
ends of the bar, thus making in all six distinct cases, as under, 
viz. — 
Case 1. When the elastic bar is held firmly by one end, the other 
end being free. 
Case 2. When both ends are free. 
Case 3. When both ends are held fast. 
Case 4. When one end is free, the other touched. 
Case 5. When one end is held, the other touched. 
Case 6. When both ends are touched. 
Now, when the end of the bar is held fast, both the ordinate and 
its derivative must be zero for that end ; when the end is free, the 
second and third derivatives must be zeroes there ; and lastly, 
when the end is touched without angular tension, the ordinate and 
its second derivative must each be zero ; and therefore the pecu- 
liarities of the equations applicable to the various cases are easily 
found. 
Case I. 
In the first and most familiar case, when the elastic bar is held 
firmly by one end, it is -easily shown that if X be the entire length, 
