176 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
We can use various values of n,, but the best is when n,2k (m™ — 1)/(m? — 1) =1, which 
gives for the magnifying power 
Mi . |m?—1 m** 
Vi a V—_— :— 
ie 2IVEKY mM m*—1 
x can not be less than 1, and m is usually much greater, so that the radical never differs 
much from unity; it can therefore be neglected, and we see that the maximum value of 
V is independent of the number of levers used, if we give n, its best values. If we use 
only one lever, n,7=1/k. This is not always practicable; for instance, for the Bosch- 
Omori instrument, 1/k = 220,000; .*. n, = 470; and since 1, =75 em., 1, becomes 0.15 cm. 
On the other hand, we can determine the best numbers of levers to use by determining 
the maximum value of V for variations of x in equation (94). This gives 
(95) 




2 21, inne 
en log zea nk V (max) = — Mil (m? — 1) (96) 
2 log m nyk 2mIVkV mi? — 1— 2k 
For the Bosch-Omori instrument, n,"k = 345, about, and with m= 10, x becomes 2.17, and 
the maximum value of V is 78. If we omit n,’k and 1 in comparison with m’, the above 
expressions become 
; “| m 1 oe r M ll (97 
= lo = lo - V (maz) = ——— 
_ 2 log m Bagh 2logm ~ zk eae) 2IVk ) 

k and l are not independent; replace k by its value, /’/J. The moment of inertia, J’, 
of each lever is principally that of the long arm, as the short arm counts but little; if 
we double the length of the lever, we must at least quadruple its mass to keep it strong 
enough; we may therefore suppose its moment of inertia equal to pl‘; introducing this 
into the values of*x and of V(maz) we get 
Pe i eee V (max) = Ml 
~ 2logm ° pl2l 21V ul 


(98) 
and we see that we get a greater multiplying power, if we use short and light levers, 
rather than a smaller number of longer and correspondingly heavier ones. , depends 
on the density and distribution of material in the levers, and should be made as small 
as possible. MI/1/ I varies proportionally with 1 M, but very little with J, if lis several 
times as large as the radius of gyration of the pendulum about its center of gravity; 
therefore V (maz) can be increased by increasing M, rather than by increasing 1. 
We have not considered the solid friction of the marking point, which, as has been 
shown on page 171, increases the minimum acceleration which can be registered in the 
proportion of the multiplying power of the levers, and is in general so great that it exerts 
a controlling influence over the possible magnifying power of the instrument. The 
investigation, therefore, does not apply directly to seismographs with mechanical regis- 
tration, but would apply to instruments of the same form if direct photographic registra- 
tion, as in the Milne instrument, were used at the end of the last lever. 
This suggests a method of optical registration by which very high magnification can 
be obtained without placing the recording paper far from the instrument. In the usual 
optical method the light is reflected directly from a mirror carried by the pendulum; but 
if the mirror is carried on the axle of a magnifying lever, the angle thru which it turns 
can be increased very greatly (fig. 50). The magnifying power becomes 
a 2 dé! _ 2 Mind 
LO = I+n,7I' 

(99) 
