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iron and steel wires was the highest they would hear without 
permanent change of length. 
The writer has long intended to continue these investi- 
gations, but hitherto time has failed him, and he now only 
publishes them as he sees from an abstract of the Pro. 
Eoyal Society, January 26th, that Mr. H. Tomlimson, B.A., 
has been experimenting in the same direction. 
Dr. Bottomley pointed out an analogy between the circle 
and the logarithmic spiral, and showed how some properties 
of the former curve could be derived from the latter. 
“On the Projection of a Solid on Three Coordinate Planes,” 
by James Bottomley, D.Sc., B.A., F.C.S. 
Take a solid of any form and let sections be made by 
parallel planes fixed in the solid. Let L be the longest axis 
of the solid which is perpendicular to these planes. This 
axis and these planes we may call the primitive axis and 
the primitive planes. 
Let the primitive axis make^with the axes of x, y , 0 angles 
a, j3, y respectively. Consider a section whose area is A x . 
If this be projected on the plane xy the area of the projec- 
tion will be A^osy. This projection we may denote by A gl . 
So in like manner the projections on the other planes will 
be AiCOSj 3 and Aicosa, and may be denoted by 4*1* A xl . 
The projections of a second area A 2 will be A 2 cosy, A 2 cosj3, 
A 2 cosa, and may be denoted by A z2 , A. y2 , A x2 . Let the 
remaining planes be projected in the same manner. Then 
by addition we shall have 
A z i + A z2 + . . . A zn = cosy(Ai + A a + . . . An) (1) 
A yX + A y2 + . . . A yw = cos/3(A 1 + A a + . . . An) (2) 
A*! + A x2 + • • . A xn = cosa(Ai + A 2 + ... An) (3) 
