223 
of functions, nor do I despair of its leading to the coveted 
solutions. 
I have been asked for my proof that P + Q must be sym- 
metrical in the £S, if P 5 + Q 5 is so, page 133. I say that if 
P 5 + Q 5 is symmetrical, P 5 cannot have more than two values 
by the permutations of the z’s, viz., P 5 and Q 5 . If P + Q is 
not symmetrical, P will have more than two values by such 
permutations, and every substitution among the z’s which 
turns P into P x or P 2 will turn PPPPP into P^P^Pi or 
P 2 P 2 P 2 P 2 P 2 , i.e. P 5 will have more than two values. 
“Note on the Resistance of Fluids/’ by Dr. J. P. 
Joule, F.R.S., President. Received by the Editor, July 
21st, 1868. 
Coulomb appears to have been the first who pointed out 
clearly the two sources of resistance to bodies moving in 
fluids, the first varying in the simple ratio, the second in 
the duplicate ratio of the velocity. He found the simple 
ratio to prevail only at the very slow velocity of not more 
them half an inch per second. I have long had in view 
experiments for the purpose of further illustrating this 
subject, and possess an apparatus which, although con- 
structed so long ago as 1849, I have not experimented with 
before the last few weeks. It consists of two vertical discs 
of turned and polished steel, each 18 inches in diameter, 
which revolve between corrugated iron plates at about one 
quarter of an inch distance. The axle passes through a 
leather collar, and the weight of the discs, which is con- 
siderable, is supported by friction wheels. Revolution is 
effected by the agency of weights and pulleys. A train of 
wheels in connection with the axis, by which a papered 
drum is moved under a pen worked by a pendulum, fur- 
nished the means of measuring the velocity. The following 
are the results at which I have already arrived. They are 
referred to one disc : — 
