84 
To find the basis of the forms of P a, 3, y We need the recursion formule, 
tee ies } d 
which may be verified by computation. 
baw. = Ps, 70-1 y, 66 
WS Aa 6,B,y+ See Ze Zs Po— 12, B,Y 
LE ff | Zz Po_=18'6 y= Po en y heed 
Pe aig Lees Ae Py a” gly ae eee Tay 
oe Pa, B=18.9 = Pa, B+6,7 
mM 
Se 
WW: 
Rg 
| 
mM 
N 
NS 
642 AA BPs pg y—12 
~TTA Pe 2 yg Po, 2 Bee 
II 28 -> 28 28 Po, p, »—TIXP>xX' Pa—e, s—6, y_6_-N 
Pe 19.8 99 G14 ee ee Fe Pe oe BANG, y ee 
Pa +12, 8412 > +412. 
Making y =o in 8, 9, 10, we get recursion formulz for Pa, ;3. 
By a repeated and successive application of the formule 8, 9, 10 to any 
Pa, 3, y, it is expressed as a rational integral function of forms Pa’, 3’, y’ whose 
greatest index is 18, and therefore finite in number, and which is therefore the 
basis of the system. 
I will add, however, that by a somewhat tedious reduction it can be shown 
that the system can be pares as rational functions of 
S 6 6 6 #7 r, r dy 
Zo, > 28 28, > 78 26 28, Z, Z, Zz Zy Pos Piss, Pe3,3, Pe,9,3, 
Rate oF DECREASE OF THE INTENSITY OF SouNDS WITH TIME OF PROPOGA- 
TION. By A. Witmer Durr. 
fAbstract. ] 
PART I.—THEORETICAL. 
The intensity of sounds spreading in spherical waves from a source 
would, if no part of the energy of vibration were lost in the passage, vary 
inversely as the square of the distance. But it is certain that a consid- 
erable proportion of the sound energy must in every second be converted 
into heat, though no attempt seems to have been made to determine 
experimentally what proportion this is of the whole. The transformation 
of energy of vibration into heat energy takes place in three ways. In the 
