Abstract The present paper is confined to an investigation of the oil buffer and the limit switch. Part I is a short description of these and allied devices. Part II gives of a theory of the oil buffer, developed by the author, that part which bears on the subject matter of this paper, namely the complete calculation of the properties of a given buffer if struck by a free weight at a given speed. It is shown that this theory applies when a buffer is suitable for service with an elevator. On the other hand it is a fact, not as yet fully recognized, that a buffer satisfactory when engaged by a free weight may be wholly objectionable if the same weight instead of being free is the car or counterweight of an elevator. In Part III a very useful and simple concept is introduced under the name of “the equivalent system,” by means of which the problems in Part IV are quickly disposed of with no greater difficulties than the application of first principles. Part IV is divided into four sections. The first is an investigation of the stop of an elevator when a limit switch is opened. It is found that the limit switch alone is inadequate as an emergency terminal stopping device unless the top and bottom clearances are exorbitant. The second is an investigation of the stop of an elevator when brought about by the action of the buffer alone. It is shown that an intimate relation exists between the buffer and the particular elevator it is to serve and two requirements are deduced which a suitable buffer must satisfy. One of these relates to the force it must exert on the member engaging it and the other to the stroke. It is further shown that a well-designed buffer is always an excellent terminal stopping device for the descending member of the elevator but not, in general, for the ascending member, unless the buffer stroke and the top clearance are considerable. This part also includes a discussion of the present buffer testing in the field and in particular of what is involved if such tests are conduced at governor tripping speed — as some codes now specify — with buffers of the usual stroke. Having to abandon the accepted notion that in the oil buffer and in the limit switch there are two independent emergency terminal stopping devices, each capable of bringing the elevator to a safe stop within the usual clearances, there is next investigated the case when both of these devices become simultaneously effective. The results are gratifying although they plainly indicate that the coöperation of both devices is necessary. They further show that the location of the limit switches is a matter of major importance. The third section investigates the stop of the elevator with buffer and limit switch coöperating in the usual sequence of practice. One of the outstanding results of Part IV is the proof that the required amount of top clearance can be readily calculated for any given emergency. Vice-versa, it is shown that a given top clearance provides for certain definite emergencies only. The author shows how to calculate the required top clearance and, upon a consideration of the accumulated experience, specifies the emergency upon which the calculation is based. It is shown in Part IV that the selection of the buffer stroke not only depends on the speed but also on the particular details of the elevator and location of the limit switches. The author further discusses which of a number of buffers of the same stroke but with different graduations is suitable for service with a given elevator. Since the emergency stops at high speed are generally associated with slack hoist rope, Part V develops the formulas for determining the stresses upon the subsequent fall of the ascending member to pick up the slack. In it will be found the reason why present methods of field testing are so frequently accompanied by a failure of the shaft of the machine and often make a re-alignment necessary.