SCIENCE. 



[Vol. XI. No. 264 



that the float gradually rides higher and higher on the water as 

 more and more chain passes over the wheel. All mechanical ar- 

 rangements of fusee or other expedient to secure uniform flotation 

 are entirely unnecessary, since the variable flotation in this case fol- 

 lows a well-defined linear law, and is perfectly compensated for by 

 a proper choice of the diameter of the wheel taken in connection 

 with the number of divisions into which it is graduated ; that is, we 

 ■do not make the divisions on the disk to correspond to the amount 

 of chain passing over the wheel, but to the actual rise of water in 

 the tube, regardless of what the former may have been. However, 

 since we wish to record each five-hundredth of an inch of rainfall, 

 the rise of water in the tube necessary to cause the wheel to make 



just one revolution must be some multiple of five-tenths of an inch, 

 -as the pins in the graduated disk must be equally distant through- 

 out, and five-tenths of an inch of water in the tube correspond 

 to five-hundredths of an inch of rainfall. 



Moreover, tlie outer circumference of the wheel must likewise be 

 some multiple of the length of the links of the chain in order that 

 the teeth may be equally distant. The dimensions of the wheel and 

 other parts to fit any particular chain are therefore chosen under 

 ■certain limitations, but are easily found as follows : — 

 Let D = the diameter of the tube C. 



" d = " " " ■' float. 



" w = weight of unit length of chain. 



" w = " " " volume of water. 



" R = radius of the wheel on its pitch line. 



" m = number of pins to be placed in the wheel. 



" 71 = " " teeth in wheel. 



" / = length of links of chain from centre to centre. 



" /i„ = depth of water when float is just supported at bottom. 



"" A = any depth of water to be measured. 



" L = length of chain passing over wheel while the float rises 

 on this water. 



"' A' = amount the float rises out of the water in coming to 

 this position. 



In its upper position the float displaces less water than when just 

 supported at the bottom, the difference being a volume, 



/'', 



4 



and the weight of this volume is equal to twice the weight of chain 



passing over the wheel in reaching its upper position, or 



jrrf- StuL 



w'/i' = iwL, and /;' = : 



hence, while the float has risen a distance L, the surface of the 

 water has risen a distance L—h', and its height from the bottom of 

 the tube is L—h' +h„ ; but the gauge is so made that the true rain- 

 fall is measured not from the surface around the float, but from the 

 surface the water would assume were the float entirely removed. 



The volume of water occupying the annular space around the 

 float is 



— {D''-dr-){h„-h'). 

 4 

 When the float is removed, this volume may be considered as 

 spreading itself out in a layer of thickness t, given by the ex- 

 pression 



■kD- tt 



/ = — {D--d-)(h„-Ii). 



4 4 



But the former thickness of the annular volume of water was 

 h^—Ji! ; hence, on removing the float, the surface of the water will 

 fall a distance {h„—k')—t, which will be found to be 

 d' 



— (Ii^-Ji). 

 D- 

 The true amount of rainfall is therefore found, after reduction, 

 to be 



( 8k/ r I I 1 ) D''—d- 



h = l\i \^ h^. 



( TTW' [d' £>■ ) ) D- 

 The last term is the amount of rainfall in true measure that must 

 collect before the float begins to be lifted, and is the number on the 

 graduated disk that must come opposite the inde.x-point when the 

 float touches the bottom of the gauge. 



To find the radius of the wheel, we will consider one complete 

 revolution and the rise of water in the tube necessary to pro- 

 duce this amount of motion. 



For this we must have 



o.5;« = 2-Ri I I 



but ittR = nl 



-dJ f I 

 TTW' (_ d'^ 



inl 



( 8w r I I ~1 



3 I I I 



{ TTW' {d' D-] 



f M 



or m = n 



d'-} 



for the quantities are all known but m, n, and d. 

 This equation may also be written 

 nb 



d"- = , 



an — m 

 and from these the value of R is found as follows : a trial value of 

 dis assumed of approximately its desired final value. With this 

 in the first of the last two equations, a few values of m are computed, 

 using such consecutive values of » (the number of teeth) as would 

 correspond to a wheel of reasonable size. In all probability, none 

 of these values of m will be whole numbers, but some one of them 

 will doubtless be very nearly a whole number. Taking the inte- 

 gral part of this and the corresponding value of n, the final value of 

 d is computed from the last equation. With the same value of 

 H, the radius of the wheel is given by the expression 



R = —, 

 and all the elements of the gauge are completely determined. 



C. F. M.'iRVIN. 

 Washington, D.C., Feb. 13. 



