384 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



as to render necessary the consideration of those changes that have 

 gone on during the immense periods of geology. 



Consequently the equations that will be deduced in the follow- 

 ing pages should, strictly speaking, include a quantity ± e, where e 

 refers to the uncertainty peculiar to the nature of their average 

 values; but for simplicity this will be omitted. 



Before I proceed to give the promised theorems, I present first 

 the notations that will be employed as follows: 



/, the time in minutes counted from the beginning of the year. 



T — 525949, the duration of a mean solar year in minutes. 



q' the quantity of heat that, at the time t, enters a unit surface 

 in a unit of time, at any given point of the earth's surface or 

 of the atmosphere, or which in a certain sense may be said to 

 pass through the unit surface. 



q" the quantity of heat that passes outward through the elemen- 

 tary unit surface, or that flows through it in the opposite 

 direction. 



qt 12 and q"t l2 the quantities of heat that enter or leave the unit 

 surface in the intervals of time between t x and t 2 or briefly q x 

 and q r when the interval of time t 12 between t t and t 2 is indi- 

 cated by r. 



q the quantity of heat entering the unit surface in the unit of time 

 at the upper boundary of the atmosphere. 



'/'i,2 — Qr an d Q the corresponding quantities for the interval t l2 

 and for the whole year. 



c h Qh,2 — Qr an( l Q the corresponding quantities of heat leaving 

 the unit surface at the upper limit of the atmosphere. 



q' and q" the quantities of heat entering and leaving a closed sur- 

 face of definite extent in a unit of time at the moment t. 



q and q the corresponding values for the boundary of the atmos- 

 phere, that is to say, for a spherical surface enclosing the 

 whole atmosphere and earth. 



tyi,2> tyi,2> tyi,2> fyi2 or briefly q-r, q T , q T , q T the corresponding values 



for the interval from t t to t 2 . 

 0/ Q," Qand 0, the corresponding values for the whole year, i.e., for 



t 2 -t,= T. 



I l n n — — = 



c\a- qb< qa' q&- q a < q&> qo, etc., the corresponding quantities for definite 

 portions a, b, of the above-named surface or for the boundary 

 of the atmosphere for the unit of time. 



q T q^ Ti etc., the corresponding quantities for the interval t l2 



